Albanese Torsors and the Elementary Obstruction Than a Mere Comparison Between the Orders of Two Classes

arXiv:math/0611284v2 [math.AG] 16 Aug 2007 rirr field arbitrary a aual soit ihi w neeswihgv mea a give which integers two it with associate naturally can rv npriua that particular in prove retraction. Galois-equivariant a fdimension of of condition enda h raetcmo iio ftedgeso h cl the of degrees the of divisor common greatest the as defined rra lsdfil Term321,o if or 3.2.1), (Theorem field closed real or hc steodro h ls of class the of order the is which n nie ftrosudrtori. Brauer, under by torsors raised of numbe the first indices a On algebras, and by simple investigated Lichtenbaum. central been Shafarevich, for has Cassels, problem equal Ogg, be [16]), to (see fail can index the iadvreyof variety Picard h ne.I h lsia iuto where situation classical the In index. the index ne eiaeinvreissc htteeeit morph a exists there that such varieties semi-abelian under torsor foei rmrl neetdi opeevreis Indee varieties. complete in interested primarily subset is one if of sauieslojc mn opim from morphisms among object universal a is obviously of period 0 n tde n[]aohrgnrlosrcint h existe the to obstruction general another [5] in studied and lmnayosrcin aey eoigby denoting Namely, obstruction. elementary cceo degree of -cycle h hm ftepeetppri htteetoobstructions two these that is paper present the of theme The If Date Let ut eeal,if generally, Quite X α X if X ⊗ ln.Frisac,if instance, For alone. I X etme 6 06 eie nAgs 5 2007. 15, August on revised 2006; 26, September : X U U k P NABNS OSR N H LMNAYOBSTRUCTION ELEMENTARY THE AND TORSORS ALBANESE ON ( sapicplhmgnossae rtro,udrasemi-ab a under torsor, or space, homogeneous principal a is X gen k o eoea rirr mohvreyover variety smooth arbitrary an denote now ⊆ → facmlt variety complete a of h lmnayosrcini adt aihi n nyi t if only and if vanish to said is obstruction elementary the , P P ) ( gen X ( saoe n h period the and above, as X Alb X ≤ ( ) ssalenough. small is k 1 = ) X X/k 1 stespeu of supremum the as 1 Term22.When 2.2). (Theorem 1 = ) 1 X ntesneo er,w r bet bansrne statemen stronger obtain to able are we Serre, of sense the in oeiton exist to Hence . sfiie(hoe ..) egv eea plctoso th of applications several give we 3.3.1); (Theorem finite is (since U see qiaett h aihn fteeeetr obstru elementary the of vanishing the to equivalent even is ⊆ P gen X Alb X P ( ( soe es,then dense, open is X X X steSvr-rurvreyatce oacnrlsml al simple central a to attached variety Severi-Brauer the is X/k 1 X ti eesr that necessary is it , = ) 1 = ) fe rvdsvlal nomto hc sntgvnb th by given not is which information valuable provides often sjs on) whereas point), a just is X P P P nteGli ooooygroup cohomology Galois the in (Alb ( U ( k LVE WITTENBERG OLIVIER mle h aihn fteeeetr btuto,oe a over obstruction, elementary the of vanishing the implies X ) sa is 1. ) when k X/k 1 A stespeu of supremum the as Introduction sanme edadteTt-hfrvc ru fthe of group Tate-Shafarevich the and field number a is sa bla ait,teetn owihtepro and period the which to extent the variety, abelian an is k X p ) eid foe aite unott erlvn even relevant be to out turn varieties open of Periods . ai ed elcoe ed ubrfil,o field a or field, number a field, closed real a field, -adic eaal lsr of closure separable a U otrosudrsm-bla aite:teAlbanese the varieties: semi-abelian under torsors to P 1 agsoe l es pnsbesof subsets open dense all over ranges ( P U k gen ) ntncsaiycmlt) n a en the define may One complete). necessarily (not divides ( X h eido ml nuhdneopen dense enough small a of period the d ism sapriua aeo h td fperiods of study the of case particular a is P 1 = ) c of nce te ad h elkonperiod-index well-known the hand, other ueo t otiilt:teindex the nontriviality: its of sure ( sdpit of points osed U X fatos nldn agadTate and Lang including authors, of r ) I P olo-hln n ascdefined Sansuc and Colliot-Thélène . → ( a esont qa h exponent the equal to shown be can X ( r eytgtyrltd eshall We related. tightly very are Y 0 la variety elian Y ) cce fdegree of -cycles ) hs fw en h generic the define we if Thus, . k einclusion he swspoe ySre there Serre, by proven was As . when n by and H 1 ( ,A k, X Y s eut n§.O the On §3. in results ese n h period the and , k ) agsoe l torsors all over ranges s otntby the notably, Most ts. h eiddivides period The . ( X A to if ction k ) vrafield a over ⋆ h ucinfield function the ⊂ 1 h so-called the , k gebra X ( k X hnfra for then , sa is ) ⋆ period e α admits P k p then , I one , -adic ( ( X X ) ) n , , 2 OLIVIER WITTENBERG other hand, if k is a field of dimension ≤ 1, we prove that the elementary obstruction always vanishes (Theorem 3.4.1), while it is well known that P (X) (and therefore Pgen(X)) may be greater than 1. The elementary obstruction to the existence of 0-cycles of degree 1 on X is known to vanish if and ⋆ only if the Yoneda equivalence class of a certain 2-extension e(X) of Pic(X ⊗k k) by k in the category of discrete Galois modules is trivial (see [5, Proposition 2.2.4]). One is naturally led to consider an analogous 2-extension E(X) of the relative Picard functor PicX/k by the multiplicative group Gm; it is a straightforward generalisation of the elementary obstruction. We devote the last section of this paper to showing that the 2-extension E(X) contains enough information to reconstruct, in a strong sense, the 1 Albanese torsor AlbU/k for any open U ⊆ X (Theorem 4.2.3). More precisely, for any open U ⊆ X, the M 2-extension E(X) gives rise, by pullback, to a 2-extension E (U) of the Picard 1-motive of U by Gm. 1 Theorem 4.2.3 essentially states that AlbU/k is the variety which parametrises, in an appropriate sense, “Yoneda trivialisations” of the 2-extension EM(U). In particular, the Yoneda equivalence class of the M 1 ∅ 2-extension E (U) is trivial if and only if AlbU/k(k) 6= , that is, if and only if P (U) = 1. From this we deduce, without any assumption on the field k, that the Yoneda equivalence class of E(X) is trivial if and only if Pgen(X) = 1 (Corollary 4.2.4). This is of course stronger than our previous assertion that the elementary obstruction vanishes if Pgen(X) = 1. M 1 Producing trivialisations of the 2-extension E (U) from rational points of AlbU/k (or from rational points of U) is not particularly difficult (see Proposition 4.1.6 and the proof of Corollary 4.2.4); it is M 1 however much less clear how to convert trivialisations of E (U) into rational points of AlbU/k. One of the key tools in the proof of Theorem 4.2.3 is a result which might be of some independent interest: we exhibit an explicit Poincaré sheaf on any abelian variety (Theorem 4.3.2).

Acknowledgements. This work grew out of an attempt to answer the questions raised by Borovoi, Colliot-Thélène, and Skorobogatov in [1], and to get a better understanding of a result of Skoroboga- tov [35, Proposition 2.1]. In response to my queries about automorphisms of n-extensions, Shoham Shamir very kindly directed me to the papers of Retakh [27] and of Neeman and Retakh [23], and explained to me a variant of Lemma 4.4.1 below. I am grateful to Dennis Eriksson for our many stim- ulating discussions about Albanese torsors and generic periods; it should be noted that generic periods ﬁrst appeared in [8]. Finally I am pleased to thank Jean-Louis Colliot-Thélène for his constant interest and encouragement, and Joost van Hamel for a number of useful comments and for pointing out the connections between §4 and [37].

Notation. Let k be a field. A variety over k is by definition a k-scheme of finite type. The Néron- Severi group NS(X) of a smooth k-variety X is the quotient of Pic(X) by the subgroup of all classes which are algebraically equivalent to 0 over k. If A0 is an algebraic group over k and A1 is a k-torsor under A0, we shall generally denote by An the n-fold contracted product of A1 with itself under A0 (so that the class of An in the Galois cohomology group H1(k, A0) is n times the class of A1). If A is a semi-abelian variety over k (that is, an extension of an abelian variety by a torus), we denote by TA the largest k-torus contained in A (so that A/TA is an abelian variety). Let A be an abelian category. We denote by C(A ) the category of complexes of objects of A . If f is a morphism in C(A ), we also denote by C(f) the mapping cone of f ([14, §1.4]); the context will make it clear which meaning is intended. n For n ∈ N and A, B ∈ A (resp. A, B ∈ C(A )), we denote by ExtA (A, B) the corresponding Ext (resp. hyperext) group. If A is the category of discrete modules over a ring R (resp. over a profinite n n n group Γ), we write ExtR(A, B) (resp. ExtΓ(A, B)) for ExtA (A, B). Finally, by a 1-motive we shall always mean a Deligne 1-motive, i.e., a 1-motive as defined in [26]. ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 3

2. The order of the elementary obstruction Let k be a field and X be a geometrically integral variety over k. The Albanese variety and the Albanese torsor of X over k, if they exist, are a semi-abelian variety 0 1 0 1 AlbX/k over k and a k-torsor AlbX/k under AlbX/k, endowed with a k-morphism uX : X → AlbX/k. They are characterised by the following universal property: for any semi-abelian variety A0 over k, any torsor A1 under A0, and any k-morphism m: X → A1, there exists a unique k-morphism of varieties 1 1 1 1 f : AlbX/k → A such that f ◦ uX = m, and there exists a unique k-morphism of algebraic groups 0 0 0 1 0 f : AlbX/k → A such that f is f -equivariant. Obviously, the Albanese variety, the Albanese torsor, and the morphism uX are unique up to a unique isomorphism. Their existence was proven by Serre [30] in case k is algebraically closed. By Galois descent it follows that they exist as soon as k is perfect (see [8, Theorem 2.1] and [32, Ch. V, §4]). As explained in the appendix to this paper, Serre’s arguments can be made to work over separably closed fields (Theorem A.1). Galois descent then implies that the Albanese variety, the Albanese torsor, and the morphism uX always exist, without any assumption on k. We henceforth assume that X is smooth over k. If X is proper over k, the Albanese variety just defined coincides with the classical Albanese variety. 0 In other words, the semi-abelian variety AlbX/k is then an abelian variety, and as such, it is canonically 0 0 dual to the Picard variety PicX/k,red of X over k (see [12, Théorème 3.3]; note that PicX/k,red is an abelian variety by [12, Corollaire 3.2]). 1 The period of X over k, denoted P (X), is by definition the order of the class of the torsor AlbX/k in 1 0 the Galois cohomology group H (k, AlbX/k). The generic period Pgen(X) of X over k is the supremum of P (U) when U ranges over all dense open subsets of X. The index I(X) of X over k is the lowest positive degree of a 0-cycle on X.

Proposition 2.1 — The period, the generic period, and the index satisfy the divisibility relations

P (X) | Pgen(X) | I(X).

Proof — It suﬃces to prove that P (X) divides I(X). Indeed, by applying this result to all dense open subsets U ⊆ X, one deduces that Pgen(X) divides I(X), thanks to the well-known fact that I(U) = I(X) for any dense open subset U ⊆ X (see [3, p. 599]). For P (X) to divide I(X), it suﬃces n ∅ n that AlbX/k(k) 6= for each n > 0 such that X contains a closed point of degree n, where AlbX/k 1 n n 1 n denotes the n-fold contracted product of the torsor AlbX/k. On composing (uX ) : X → (AlbX/k) 1 n n n n with the projection (AlbX/k) → AlbX/k, we obtain a morphism X → AlbX/k which is symmetric, and which therefore factors through the n-fold symmetric power X(n) of X ([21, Proposition 3.1]). It now only remains to prove that a closed point of degree n on X gives rise to a rational point of X(n). This is clear if k is perfect; in general it follows from the existence of the Grothendieck-Deligne norm map from the Hilbert scheme of points of degree n on X to X(n) (see [6, p. 184]).

Let k denote a separable closure of k and let Γ = Gal(k/k). The elementary obstruction (to the 1 ⋆ ⋆ ⋆ existence of a 0-cycle of degree 1 on X over k) is the class ob(X) ∈ ExtΓ(k(X) /k , k ) of the exact sequence of discrete Γ-modules ⋆ ⋆ 0 k k(X)⋆ k(X)⋆/k 0, 4 OLIVIER WITTENBERG where k(X) denotes the function ﬁeld of X ⊗k k. It is an easy consequence of Hilbert’s Theorem 90 that the existence of a 0-cycle of degree 1 on X forces ob(X) to vanish (see [5, Proposition 2.2.2]). More generally, the same argument shows that the order of ob(X) divides I(X). Our main result in this section is the following:

Theorem 2.2 — Let X be a smooth proper geometrically integral variety over a ﬁeld k. The order of ob(X) divides Pgen(X).

In particular, the elementary obstruction to the existence of a 0-cycle of degree 1 on X is even an obstruction to the generic period of X being equal to 1. The integers P (X), Pgen(X), I(X), and the order of ob(X) satisfy no further divisibility relations than those given by Proposition 2.1 and Theorem 2.2, as we now show with a few examples. First of all, the order of ob(X) does not divide P (X) in general: any curve of genus 0 without rational points has period 1 but ob(X) 6= 0. Conversely, the order of ob(X) is not necessarily divisible by P (X): if X is a curve of genus 1 without rational points over a field of dimension ≤ 1 (for instance the plane cubic x3 + ty3 + t2z3 = 0 over C((t))), then clearly P (X) > 1, but Theorem 3.4.1 below implies that ob(X) = 0. The period and the generic period differ in the case of a curve of genus 0 without rational points. Examples where the generic period and the index do not coincide are given by Severi-Brauer varieties attached to central simple algebras for which the exponent and the index are not equal. One can even give examples of varieties which satisfy Pgen(X) = 1 and I(X) > 1. According to Theorem 3.2.1 below, it suffices to exhibit a geometrically integral variety X, over a p-adic field, such that ob(X) = 0 and I(X) > 1. As explained in [1, §2.2, Remark 2], any smooth cubic projective hypersurface of dimension at least 3 without rational points over a p-adic field satisfies these conditions, thanks to a theorem of Coray and to Max Noether’s theorem. Using Theorem 3.3.1 below, it is also possible, although even more involved, to give examples of (geometrically) rational surfaces over Q for which Pgen(X) = 1 and I(X) > 1. However such examples do not exist in dimension 1 (one can show that Pgen(X)= I(X) for any curve X over any field k). The remaining of this section will be devoted to the proof of Theorem 2.2. We refer the reader to §3 for applications of arithmetical interest.

0 Proof of Theorem 2.2 — For any dense open subset U ⊆ X, let TU/k be the k-torus whose character ⋆ ⋆ group is k[U] /k , where k[U] denotes the ring of regular functions on U ⊗k k. There exist a k-torsor 1 0 1 TU/k under TU/k and a k-morphism U → TU/k which satisfy the following universal property: for any 0 1 0 1 1 k-torus R and any k-torsor R under R , any k-morphism U → R factors uniquely through TU/k (see [34, Lemma 2.4.4]). The exact sequence of discrete Γ-modules

⋆ ⋆ 0 k k[U]⋆ k[U]⋆/k 0

1 ∅ U splits if and only if TU/k(k) 6= (see [34, Lemma 2.4.4]). Similarly, for any n ≥ 1, if denotes the n 1 set of all dense open subsets of X and TU/k denotes the n-fold contracted product of TU/k as a torsor 0 n under T , then lim (T (k)) 6= ∅ if and only if there exists a family (r ) U of Γ-equivariant U/k ←−U∈U U/k U U∈ ⋆ ⋆ n ⋆ homomorphisms rU : k[U] → k such that rU (x) = x for all x ∈ k and all U, and such that rU ⋆ and rV coincide on k[U] for all U and all V ⊆ U. Now this condition holds if and only if there exists ⋆ ⋆ a Γ-equivariant homomorphism r : k(X)⋆ → k such that r(x) = xn for all x ∈ k . This in turn is ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 5 equivalent to the order of ob(X) dividing n. We are thus reduced to proving that lim (Tn (k)) 6= ∅ ←−U∈U U/k if n = Pgen(X). n n The universal property of the Albanese torsor provides us with a k-morphism AlbU/k → TU/k for each U ∈ U , functorially in U. Hence there is a map lim (Albn (k)) → lim (Tn (k)). In ←−U∈U U/k ←−U∈U U/k particular we need only prove that lim (Albn (k)) 6= ∅ as soon as P (X) divides n, and this is ←−U∈U U/k gen achieved by the following proposition.

Proposition 2.3 — Let X be a smooth proper geometrically integral variety over k and let n ≥ 1. U n ∅ U Denote by the set of all dense open subsets of X. If AlbU/k(k) 6= for all U ∈ , then lim (Albn (k)) 6= ∅. ←−U∈U U/k

Proof — We begin with two lemmas.

Lemma 2.4 — Let U ⊆ V ⊆ X be dense open subsets, and suppose that NS(V ⊗k k) = 0. Then there is an exact sequence of k-group schemes

0 m 0 0 Q AlbU/k AlbV/k 0 where Q is a quasi-trivial k-torus and m is induced by the inclusion U ⊆ V .

(Recall that a k-torus is said to be quasi-trivial if it is isomorphic to a product of Weil restrictions of scalars of Gm from ﬁnite separable extensions of k down to k.)

0 0 0 Proof — For any dense open subset W ⊆ X, the kernel of the natural map AlbW/k → AlbX/k is TAlbW/k 0 0 (recall that TAlbW/k denotes the largest k-torus contained in AlbW/k), and the group of characters of 0 this torus identiﬁes with the group DivX\W (X ⊗k k) of divisors on X ⊗k k which are algebraically equivalent to 0 and supported on X \ W (see Theorem A.4). As a consequence, we need only prove that 0 0 the map TAlbU/k → TAlbV/k induced by m is surjective and that its kernel is a quasi-trivial k-torus. In other words, we need only prove that the natural map

0 0 DivX\V (X ⊗k k) −→ DivX\U (X ⊗k k) is injective and that its cokernel is a permutation Γ-module. Injectivity is tautological. The hypothesis NS(V ⊗k k) = 0 implies that the cokernel is naturally isomorphic to DivV \U (V ⊗k k), which is indeed a permutation Γ-module.

Lemma 2.5 — Let U, V be dense open subsets of X such that NS((U ∪ V ) ⊗k k) = 0. For any n ≥ 0, the natural morphism n n n Alb −→ Alb × n Alb (U∩V )/k U/k Alb(U∪V )/k V/k is an isomorphism. 6 OLIVIER WITTENBERG

Proof — We may assume k = k and then n = 0. Consider the commutative diagram of algebraic groups 0 0 0

0 0 0 0 0 TAlb(U∩V )/k TAlbU/k × TAlbV/k TAlb(U∪V )/k

0 0 0 0 0 Alb(U∩V )/k AlbU/k × AlbV/k Alb(U∪V )/k

0 0 0 0 0 AlbX/k AlbX/k × AlbX/k AlbX/k ,

0 0 0 in which the leftmost (resp. rightmost) horizontal maps are the products (resp. differences) of the maps induced by the various inclusions. The statement of the lemma amounts to the exactness of the middle row. The bottom row and the columns are exact, so it suffices to prove that the top row is exact. For this it suffices to prove that the corresponding sequence of character groups is exact. By Theorem A.4, the latter identifies with

0 0 0 0 DivF ∩G(X) DivF (X) × DivG(X) DivF ∪G(X) 0, where F = X \ U, G = X \ V , the ﬁrst map sends a divisor D to (D, −D), and the second map sends (D, D′) to D + D′. Exactness in the middle is obvious. Exactness on the right follows from the hypothesis that NS(U ∪ V ) = 0.

We now return to the proof of Proposition 2.3. Let V ⊆ X be a dense open subset such that n NS(V ⊗k k) = 0 (such a V exists because NS(X ⊗k k) is finitely generated). Choose xV ∈ AlbV/k(k). Let n n F ⊂ V be an irreducible closed subset of codimension 1. The natural morphism m: Alb(V \F )/k → AlbV/k is a torsor under the k-torus Q of Lemma 2.4. Since Q is quasi-trivial, this implies that all fibres of m contain rational points, by Shapiro’s lemma and Hilbert’s Theorem 90. We can therefore choose for each n such F a point xV \F ∈ Alb(V \F )/k(k) whose image by m is xV . Now let U be an arbitrary dense open subset of V . Denote by F1,...,Fm the irreducible components of V \U which are of codimension 1 in V . Quite generally, if W is a dense open subset of X and Z ⊂ W is a closed subset of codimension at least 2, n n the natural map Alb(W \Z)/k → AlbW/k is an isomorphism (this is a consequence of [19, Lemma 3.3]). In particular, the natural morphism Albn −→ Albn is an isomorphism, as well as the U/k (V \(S1≤i≤m Fi))/k natural morphism Albn → Albn for any subset I ⊆ {1,...,m} of cardinality at least 2. (V \(T1≤i≤m Fi))/k V/k In view of these remarks, Lemma 2.5 implies that the natural morphism n n n Alb −→ Alb × n ···× n Alb U/k (V \F1)/k AlbV/k AlbV/k (V \Fm)/k n U is an isomorphism. Define xU ∈ AlbU/k(k) to be the inverse image of (xV \F1 ,...,xV \Fm ). Letting V n denote {U ∈ U ; U ⊆ V }, we have now produced a whole family x = (x ) U ∈ U (Alb (k)). U U∈ V QU∈ V U/k It is immediate that x is an element of lim (Albn (k)), hence lim (Albn (k)) 6= ∅. ←−U∈UV U/k ←−U∈U U/k ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 7

Remark 2.6 — It is not known whether the statement of Proposition 2.3 remains true, even for n = 1, n n if one replaces all occurrences of AlbU/k in it with TU/k. In other words, while the vanishing of ob(X) clearly implies the existence of a Γ-equivariant splitting of the exact sequence ⋆ ⋆ 0 k k[U]⋆ k[U]⋆/k 0 for every dense open subset U ⊆ X, it is an open question whether the converse implication holds in general.

3. Arithmetical applications and fields of dimension ≤ 1 In this section we give several applications of Theorem 2.2 when k is a p-adic field, a real closed field, or a number field. Our main result here is Theorem 3.3.1, which answers positively a question posed by Borovoi, Colliot-Thélène, and Skorobogatov in [1] about the elementary obstruction over number fields. In addition, in the three cases under consideration, we address the following deceptively simple-looking general question, which was also raised by the aforementioned authors in [1] (see [1, §2.1]; there k is assumed to have characteristic 0):

Question 3.1 — Let K/k be a field extension and X be a smooth proper geometrically integral k-variety. Let ob(X ⊗k K) denote the class of the elementary obstruction on the K-variety X ⊗k K. Does ob(X) = 0 imply ob(X ⊗k K) = 0 ? We then proceed to consider the elementary obstruction and generic periods over fields of dimension ≤ 1 (in the sense of Serre [33, II.§3]). The results we obtain for fields of dimension ≤ 1 are in marked contrast with those for p-adic or real closed fields and number fields. 3.1. Preliminaries. We first establish two general lemmas that will be used below.

Lemma 3.1.1 — Let k be a ﬁeld and X be a smooth proper geometrically integral variety over k. Let V ⊆ X be a dense open subset such that NS(V ⊗k k) = 0. Then P (V )= Pgen(X).

Proof — Let W ⊆ V be a dense open subset. We need only prove that P (V )= P (W ). By Lemma 2.4 1 0 1 0 and Hilbert’s Theorem 90, the Galois cohomology map H (k, AlbW/k) → H (k, AlbV/k) induced by the 1 1 inclusion W ⊆ V is injective. It sends the class of AlbW/k to the class of AlbV/k; therefore these two classes have the same order.

Lemma 3.1.2 — Let k be a ﬁeld and f : X 99K Y be a rational map between smooth geometrically integral varieties over k. If ob(X) = 0, then ob(Y ) = 0.

Proof — After shrinking X, we may assume f is a morphism. We denote as usual k a separable closure of k and Γ = Gal(k/k). Let ξ ∈ Y ⊗k k be the image by f of the generic point of X ⊗k k. There is a natural exact sequence of discrete Γ-modules

(3.1.1) 0 O⋆ k(Y )⋆ D 0, Y ⊗kk, ξ where D is the group of divisors on Y ⊗k k whose support contains ξ. Now D is a permutation Galois 1 O⋆ module, so that the short exact sequence (3.1.1) splits (indeed ExtΓ(D, ) = 0 by Shapiro’s Y ⊗kk, ξ lemma and Grothendieck’s Hilbert 90 theorem). Composing a Γ-equivariant retraction of the leftmost map of (3.1.1) with the evaluation map O⋆ → k(ξ)⋆ and the natural inclusion k(ξ)⋆ ⊆ k(X)⋆, we Y ⊗kk, ξ 8 OLIVIER WITTENBERG

⋆ finally obtain a Γ-equivariant map k(Y )⋆ → k(X)⋆ which induces the identity on k . The existence of a ⋆ Γ-equivariant retraction of the inclusion k ⊆ k(Y )⋆ then follows from the existence of a Γ-equivariant ⋆ retraction of the inclusion k ⊆ k(X)⋆. 3.2. Local fields. In [1, Theorem 2.7], Borovoi, Colliot-Thélène, and Skorobogatov proved that if X is a smooth proper geometrically integral variety over a p-adic field or a real closed field, then ob(X) = 0 implies P (X) = 1. This was a reformulation of a result of van Hamel [37], which itself generalised earlier work of Roquette and Lichtenbaum. It is easy to see that the implication ob(X) = 0 ⇒ P (X) = 1 fails to be an equivalence over p-adic or real closed fields in general — any curve of genus 0 without rational points furnishes a counter-example. The following theorem strengthens [1, Theorem 2.7] and supplements it with a converse.

Theorem 3.2.1 — Let k be a p-adic ﬁeld or a real closed ﬁeld and X be a smooth proper geometrically integral variety over k. Then ob(X) = 0 if and only if Pgen(X) = 1.

Proof — If Pgen(X) = 1, then ob(X) = 0 by Theorem 2.2. Now assume ob(X) = 0 and let U ⊆ X 1 ∅ be a dense open subset. We need only prove that P (U) = 1, i.e., that AlbU/k(k) 6= . The existence 99K 1 1 of a rational map X AlbU/k and the condition ob(X) = 0 imply together that ob(AlbU/k) = 0 1 ∅ by Lemma 3.1.2. From this it follows that AlbU/k(k) 6= by [1, Theorem 3.2].

Remark 3.2.2 — If k is a p-adic field, it is possible to prove that the integer denoted PsI(X) in [37] coincides with Pgen(X), as van Hamel explained to the author. Granting this, the conclusion of Theorem 3.2.1 in the case of a p-adic field is also a consequence of [37, Theorem 2] and [1, Theorem 2.5]. (Note that if k is a real closed field, the integers PsI(X) and Pgen(X) need not be equal; see the remark after the proof of [1, Theorem 2.6] for an example where Pgen(X) = 1 but PsI(X) = 2.)

We can now give an affirmative answer to Question 3.1 in case the base field is a p-adic field or a real closed field:

Corollary 3.2.3 — Let k be a p-adic field or a real closed field and X be a smooth proper geometrically integral variety over k. If ob(X) = 0, then ob(X ⊗k K) = 0 for any field extension K/k.

Proof — Let V ⊆ X be a dense open subset such that NS(V ⊗k k) = 0. Suppose ob(X) = 0. By 1 ∅ Theorem 3.2.1 we then have Pgen(X) = 1, so that in particular AlbV/k(k) 6= . Now the natural map Alb1 → Alb1 ⊗ K is an isomorphism, by Corollary A.5 (note that k is perfect). Hence V ⊗kK/K V/k k Alb1 (K) 6= ∅, or equivalently P (V ⊗ K) = 1. Let K be an algebraic closure of K. The condition V ⊗kK/K k NS(V ⊗k k) = 0 implies NS(V ⊗k K) = 0. We can therefore apply Lemma 3.1.1 to the K-variety V ⊗k K and obtain the equality Pgen(X ⊗k K) = 1. Invoking Theorem 2.2 now concludes the proof. 3.3. Number fields. Let k be a number field and X be a smooth geometrically integral variety over k. Let Ak denote the ring of adèles of k. In this context, obstructions to the existence of rational points more elaborate than the elementary obstruction have been defined. Of particular importance is the Brauer-Manin obstruction associated with the subgroup B(X) ⊆ Br(X) of locally constant algebraic 2 classes in the cohomological Brauer group Br(X) = Hét(X, Gm). This obstruction is said to vanish B ∅ B B when X(Ak) 6= , where X(Ak) denotes the set of adelic points of X which are orthogonal to (X) with respect to the Brauer-Manin pairing X(Ak) × Br(X) → Q/Z ([34, §5.2]). ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 9

Borovoi, Colliot-Thélène, and Skorobogatov proved in [1, Theorem 2.13] that ob(X) = 0 implies B ∅ ∅ X(Ak) 6= , provided X(Ak) 6= . In the remark which follows their proof, they note that the converse is known to hold by a theorem of Colliot-Thélène and Sansuc if Pic(X ⊗k k) is a free abelian group (as is the case for instance if X is geometrically unirational); see [5, Prop. 3.3.2]. They then ask whether the converse holds in general. The following theorem provides an affirmative answer to this question, modulo the widely believed conjecture that Tate-Shafarevich groups of abelian varieties over number fields are finite. It also provides a converse to a theorem of Eriksson and Scharaschkin ([8, Theorem 1.1]).

Theorem 3.3.1 — Let k be a number ﬁeld and X be a smooth proper geometrically integral variety over k. Assume that the Tate-Shafarevich group of the Picard variety of X over k is ﬁnite. Consider the following conditions: B ∅ (i) X(Ak) 6= ; (ii) ob(X) = 0; (iii) Pgen(X) = 1. ∅ Conditions (ii) and (iii) are equivalent. If X(Ak) 6= , then (i), (ii), and (iii) are equivalent.

Note that the properness assumption on X is harmless; indeed, for any dense open subset U ⊆ X, B ∅ B ∅ ∅ the conditions X(Ak) 6= and U(Ak) 6= are equivalent, since on the one hand X(Ak) 6= implies ∅ U(Ak) 6= (by Nishimura’s lemma [15, IV.6.2], the Lang-Weil estimates, and Hensel’s lemma) and on the other hand B(U)= B(X) (for a proof of this equality see [28, Lemme 6.1] or [8, Lemma 3.4]).

∅ Proof of Theorem 3.3.1 — We have (iii)⇒(ii) by Theorem 2.2, (ii)⇒(i) if X(Ak) 6= by [1, Theo- ∅ rem 2.13], and (i)⇒(iii) by [8, Theorem 1.1]. This proves Theorem 3.3.1 in case X(Ak) 6= . (The finiteness of the relevant Tate-Shafarevich group is only used for establishing (i)⇒(iii). The B ∅ B ∅ proof of [8, Theorem 1.1] in our context is as follows: X(Ak) 6= implies U(Ak) 6= (where 1 B ∅ U ⊆ X is an arbitrary dense open subset), which implies AlbU/k(Ak) 6= (thanks to the existence of 1 1 ∅ a morphism U → AlbU/k), and then it follows that AlbU/k(k) 6= (and thus P (U) = 1) by a theorem of Harari and Szamuely [13, Theorem 1.1], which relies on the finiteness of the Tate-Shafarevich group of the Picard variety of X.) Let us now prove Theorem 3.3.1 in general. As Theorem 2.2 still yields (iii)⇒(ii), we need only consider (ii)⇒(iii). Let U ⊆ X be a dense open subset such that P (U) = Pgen(X). By Lemma 3.1.2 99K 1 applied to the canonical rational map X AlbU/k, we may assume that U is a torsor under a semi- 1 1 abelian variety, after replacing U with AlbU/k and X with a smooth compactification of AlbU/k. (To ensure the existence of such a compactification, one can avoid Hironaka’s theorem on resolution of singularities; see the comment after Corollary 4.1.5.) In this case, it follows from condition (ii) that ∅ X(Ak) 6= by [1, Proposition 2.12 and Theorem 3.2], and we have already proven Theorem 3.3.1 under this assumption.

The following corollary gives a conditional affirmative answer to Question 3.1 in case the base field is a number field.

Corollary 3.3.2 — Let X be a smooth proper geometrically integral variety over a number field k. Assume that the Tate-Shafarevich group of the Picard variety of X over k is finite. If ob(X) = 0, then ob(X ⊗k K) = 0 for any field extension K/k. 10 OLIVIER WITTENBERG

Proof — Corollary 3.3.2 follows from Theorem 3.3.1 in the exact same way as Corollary 3.2.3 follows from Theorem 3.2.1. 3.4. Fields of dimension ≤ 1. Recall that a field k is said to be a field of dimension ≤ 1 if the Brauer group of every finite extension of k is trivial. For perfect fields, this is equivalent to having cohomological dimension ≤ 1 ([33, II.§3.1, Proposition 6]). Examples of fields of dimension ≤ 1 are (C1) fields ([33, II.§3.2]). Many properties of algebraic varieties over arbitrary fields have a tendency to either become trivial or to not get any easier to understand when they are specialised to perfect fields of dimension ≤ 1. Torsors under connected linear algebraic groups thus always have rational points over such fields (as proven by Steinberg, see [33, III.§2.3]), whereas torsors under abelian varieties do not (examples are easy to construct over C((t)); in particular generic periods of varieties over perfect fields of dimension ≤ 1 can be nontrivial). In view of Theorems 3.2.1 and 3.3.1, this prompts the question whether Pgen(X) = 1 might be equivalent to ob(X) = 0 over perfect fields of cohomological dimension ≤ 1. The answer is in the negative, as shown by the following theorem (whose proof does not make use of Theorem 2.2).

Theorem 3.4.1 — Let k be a ﬁeld of dimension ≤ 1. Then ob(X) = 0 for any geometrically integral variety X over k.

We note that if k is a ﬁeld of dimension ≤ 1, the exact sequence of Galois modules ⋆ ⋆ 0 k k[U]⋆ k[U]⋆/k 0

1 is clearly split for any dense open subset U ⊆ X (indeed the torsor TU/k which appears in the proof of Theorem 2.2 has a rational point since it is a torsor under a torus and k has dimension ≤ 1 [31, p. 170]). If this implied ob(X) = 0, Theorem 3.4.1 would follow immediately. See Remark 2.6.

To prove Theorem 3.4.1 we start with a variant of a well-known lemma in homological algebra originally due to Matlis ([17, Lemma 1]).

Lemma 3.4.2 — Let G be a proﬁnite group and M be a discrete G-module. Fix n ≥ 1 and assume n ExtG(Z[G/H]/I, M) = 0 for any normal open subgroup H ⊆ G and any left ideal I ⊆ Z[G/H]. Then n ExtG(B, M) = 0 for any discrete G-module B. Proof — We prove the lemma by induction on n. If the assertion is true for a given n ≥ 1, it is seen to ′ n+1 hold for n+1 by embedding M into an injective discrete G-module M and noting that ExtG (B, M)= n ′ ExtG(B, M /M). Thus we can assume n = 1, and the conclusion of the lemma is then equivalent to M being injective. Let E ⊆ F be an inclusion of discrete G-modules and f : E → M be a G-equivariant morphism. By Zorn’s lemma, there exists a sub-G-module E′ ⊆ F containing E and a G-equivariant morphism f ′ : E′ → M which extends f and which cannot be extended to a G-equivariant morphism on any larger sub-G-module of F . It only remains to be shown that F = E′. Let x ∈ F and let H ⊆ G be a normal open subgroup contained in the stabiliser of x. Let E′′ ⊆ F denote the sub-G-module of F generated by E′ and x. Then E′′/E′ is of the shape Z[G/H]/I for some left ideal I, and as a consequence 1 ′′ ′ ′′ we have ExtG(E /E , M) = 0. In particular f can be extended to a G-equivariant morphism E → M, so that E′′ = E′ and therefore x ∈ E′; hence ﬁnally F = E′.

Proof of Theorem 3.4.1 — Let k be a separable closure of k and Γ = Gal(k/k). According to [5, 2 ⋆ Proposition 2.2.4], it suﬃces to establish that ExtΓ(Pic(X ⊗k k), k ) = 0. We shall actually prove that 2 ⋆ ExtΓ(B, k ) = 0 for any discrete Γ-module B. By Lemma 3.4.2, we may assume that B = Z[Γ/H]/I, ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 11 where H ⊆ Γ is a normal open subgroup and I ⊆ Z[Γ/H] is a left ideal. It follows from the exact sequence 0 I Z[Γ/H] B 0

2 ⋆ 2 ⋆ 1 ⋆ that in order for ExtΓ(B, k ) to be trivial, it suffices that ExtΓ(Z[Γ/H], k ) and ExtΓ(I, k ) be trivial. The first of these groups is equal to the Brauer group of a finite extension of k (quite generally one n n has ExtΓ(Z[Γ/H], −) = H (H, −) for n = 0 hence for any n), therefore it is trivial. By [20, Ch. I, §0, Theorem 0.3 and Example 0.8], which we can apply because I is finitely generated as a Z-module, the second of these groups fits into an exact sequence

1 ⋆ 1 ⋆ 1 ⋆ 0 H (k, HomZ(I, k )) ExtΓ(I, k ) ExtZ(I, k ), where the subscript Z refers to the category of abelian groups. Now I is a free Z-module; hence on 1 ⋆ 1 ⋆ the one hand, the group ExtZ(I, k ) is trivial, and on the other hand, the group H (k, HomZ(I, k )) is equal to H1(k, T ) where T is a k-torus, and is therefore also trivial as k is a ﬁeld of dimension ≤ 1 (see [31, p. 170]).

In the statement of Theorem 3.4.1, the hypothesis that k be a field of dimension ≤ 1 cannot be replaced with the weaker hypothesis that k have cohomological dimension ≤ 1. In other words, there are geometrically integral varieties defined over (imperfect) fields of cohomological dimension 1 for which the elementary obstruction does not vanish. Even better, the statement of Theorem 3.4.1 turns out to be optimal:

Proposition 3.4.3 — Let k be a ﬁeld. Suppose that ob(X) = 0 for any geometrically integral variety X over k. Then k is a ﬁeld of dimension ≤ 1.

Proof — It suffices to prove that Br(K) = 0 for all finite separable extensions K/k ([33, II.§3.1, Proposition 5]). Let X be a Severi-Brauer variety over K. It is a toric K-variety (see for instance [18, Example 8.5.1]); as a consequence, the Weil restriction of scalars RK/kX is a toric k-variety. Now we ∅ have ob(RK/kX) = 0 by hypothesis, so that (RK/kX)(k) 6= (see [1, Lemma 2.1 (iv)], whose proof works for arbitrary fields) and therefore X(K) 6= ∅.

Let us now consider Question 3.1 under the assumption that k is a field of dimension ≤ 1. An affirmative answer would imply, together with Theorem 3.4.1, that ob(X ⊗k K) = 0 for any geometrically integral k-variety X and any field extension K/k. We only establish the following weaker result (whose proof does make use of Theorem 2.2).

Proposition 3.4.4 — Let k be a perfect ﬁeld of dimension ≤ 1 and X be a smooth proper geometrically integral variety over k. If P (X) = 1, then ob(X ⊗k K) = 0 for any ﬁeld extension K/k.

n n Proof — Let V ⊆ X be a dense open subset and n ≥ 0. The natural morphism AlbV/k → AlbX/k is 0 n a torsor under TAlbV/k. In particular its ﬁbre over any k-point of AlbX/k is a k-torsor under a torus, hence contains a rational point by [31, p. 170]. This proves that Pgen(X) = 1. Now the rest of the argument is the same as in the proof of Corollary 3.2.3. 12 OLIVIER WITTENBERG

4. Albanese torsors, 1-motives, and the elementary obstruction Let X be a smooth proper geometrically integral variety over a ﬁeld k. As is well known, the elementary obstruction to the existence of a 0-cycle of degree 1 on X vanishes if and only if the ⋆ 2-extension e(X) of Pic(X ⊗k k) by k deﬁned by the exact sequence of Γ-modules

⋆ ⋆ 0 k k(X) Div(X ⊗k k) Pic(X ⊗k k) 0

2 ⋆ gives rise to the trivial class in ExtΓ(Pic(X ⊗k k), k ), where k denotes a separable closure of k and Γ = Gal(k/k) ([5, Proposition 2.2.4]). More generally, the order of ob(X) is equal to the order of the 2 ⋆ class of e(X) in ExtΓ(Pic(X ⊗k k), k ). Theorem 2.2 can thus be reformulated as stating that the order of the class of e(X) divides the generic period Pgen(X); this divisibility is not an equality in general, according to Theorem 3.4.1. Deﬁning an analogous 2-extension E(X) of the relative Picard functor PicX/k by the multiplicative group Gm in the category of étale sheaves on Sm/k is a straightforward task (see §4.1 for details; Sm/k denotes the category of smooth k-schemes of ﬁnite type). One of our main results below is the following: 2 the order of the class of E(X) in Ext (PicX/k, Gm) is actually equal to Pgen(X) (see Corollary 4.2.4). This can be seen as an optimal strengthening of Theorem 2.2, since the order of the class of e(X), which is equal to the order of ob(X), clearly divides the order of the class of E(X). Our goal, however, is really to uncover a deeper connection between Albanese torsors and the elementary obstruction than a mere comparison between the orders of two classes. Let EM(U) denote the 2-extension of the Picard 1-motive of U by Gm obtained by pullback from E(X). The main theorem of this section (Theorem 4.2.3) states that by applying a certain canonical construction to the 2-extension EM(U), 1 1 one can recover the torsor AlbU/k as a sheaf on Sm/k (and hence as a variety, since AlbU/k ∈ Sm/k). 1 As mentioned in the introduction, it will follow that rational points of AlbU/k correspond naturally to “Yoneda trivialisations” of the 2-extension EM(U). In the case where U = X, a Yoneda trivialisation M 0 of E (X) is simply a zigzag of morphisms of 2-extensions of PicX/k by Gm, which starts with the 0 pullback of E(X) by the inclusion PicX/k → PicX/k, and ends with the trivial 2-extension

Id 0 0 Id 0 0 Gm Gm PicX/k PicX/k 0.

When X is a torsor under an abelian variety, Theorem 4.2.3 identifies X(k) with the set of such zigzags up to a certain homotopy relation (and similarly for any finite separable extension ℓ/k). The remainder of the paper is organised as follows. In §4.1 we define the 2-extensions E(X) and EM(U), and establish some of their properties. Then in §4.2 we describe the general construction 1 M which allows one to recover AlbU/k from E (U), state a key theorem (Theorem 4.2.2), and deduce Theorem 4.2.3 from it. Section §4.3 is devoted to an independent result about Poincaré sheaves on abelian varieties. The proof of Theorem 4.2.2 is finally given in §4.4. It rests on the contents of §4.3. Under the additional hypothesis that the field k has characteristic 0, another approach to some of the results contained in this section was considered earlier by van Hamel in [37], based on the well-known explicit description of the Weil pairing. Specifically, the first assertion of Corollary 4.2.4 below, in the case where U = X and k has characteristic 0, follows from [37, Theorem 3.6] (as well as from [35, Proposition 2.1]). See also Remark 3.2.2 and [36, Remark 1.7]. ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 13

4.1. Motivic 2-extensions. We fix once and for all a field k and a smooth proper geometrically integral variety X over k. (The variety X must be normal for Lemma 4.1.2 below to hold; but it is mostly for convenience that we moreover assume it to be smooth.) If S is a scheme, let Sm/S denote the category of smooth S-schemes of finite type and Sh(Sm/S) the category of sheaves in abelian groups on Sm/S for the étale topology. For S ∈ Sm/k and Y ∈ Sm/S, we denote by Div(Y/S) the subgroup of Div(Y ) consisting of those divisors whose support does not contain any irreducible component of any fibre of Y → S, we denote by R(Y/S)⋆ the group of invertible rational functions on Y whose divisor belongs to Div(Y/S), and we let Pic(Y/S) denote the cokernel of the natural map Pic(S) → Pic(Y ). We let Pic0(Y/S) be the subgroup of Pic(Y/S) consisting of those classes whose restriction to every geometric fibre of Y → S is algebraically equivalent to 0, and Div0(Y/S) be the inverse image of Pic0(Y/S) by the natural map Div(Y/S) → Pic(Y/S). If L ⊆ Y is a subscheme, we denote by Div(Y/S,L) the subgroup of Div(Y/S) consisting of those divisors whose ⋆ support is contained in L. For T ∈ Sm/S, the groups R(Y ×S T/T ) , Div(Y ×S T/T ), Pic(Y ×S T/T ), 0 0 Div (Y ×S T/T ), Pic (Y ×S T/T ), and Div(Y ×S T/T,L×S T ) depend functorially on T ; as a consequence ⋆ 0 0 they define presheaves on Sm/S. Let RY/S, DivY/S, PicY/S, DivY/S, PicY/S, and DivY/S,L denote the ⋆ 0 associated étale sheaves, and let RY/S,L (resp. DivY/S,L) be the inverse image of DivY/S,L by the natural ⋆ 0 map RY/S → DivY/S (resp. DivY/S → DivY/S). If Y is proper over S, then PicY/S coincides with the relative Picard functor (see [2, p. 201]).

Remark 4.1.1 — Under our assumptions on S and Y , the groups R(Y/S)⋆ and Div(Y/S) respectively coincide with the groups M(Y/S)⋆ and Div(Y/S) deﬁned by Grothendieck in [11, 20.6.1 and 21.15.2]. Note that for T ∈ Sm/S, the group of invertible rational functions on Y ×ST does not depend functorially on T , even if S = Y = Spec(k); hence the need to consider relative rational functions and relative divisors.

Lemma 4.1.2 — The natural complex of étale sheaves in abelian groups on Sm/k

⋆ (4.1.1) 0 Gm RX/k DivX/k PicX/k 0 is exact.

Proof — Only exactness at PicX/k requires a proof. It suffices to establish that for any connected T ∈ Sm/k, any t ∈ T , and any D ∈ Div(X ×k T ), there is an open subset U of T containing t such that the restriction of D to X ×k U is linearly equivalent, on X ×k U, to an element of Div(X ×k U/U). Let ξ O denote the generic point of the fibre of X ×k T → T above t, and let denote the local ring of X ×k T at ξ. The Picard group of any local ring vanishes, hence Pic(O) = 0, and in particular the image of D in Pic(O) is trivial, which means that after replacing D with D + P for some principal divisor P on X ×k T , we may assume that the support of D does not contain ξ. The set U of points u ∈ T such that the support of D does not contain the fibre of X ×k T → T above u is an open subset of T , by Chevalley’s semi-continuity theorem [10, 13.1.3]. By assumption it contains t, and the restriction of D to X ×k U is indeed an element of Div(X ×k U/U).

The complex (4.1.1) thus defines a 2-extension of the relative Picard functor PicX/k by the multiplicative group Gm in the category Sh(Sm/k). We let E(X) denote this 2-extension. Let U ⊆ X be a dense open subset. With U are naturally associated two 1-motives: the Albanese 1 1-motive M1(U) and the Picard 1-motive M (U). By definition M1(U) is the semi-abelian variety 0 1 AlbU/k and M (U) is the Cartier dual of M1(U) (see [25, §2.1]; for the definition of Cartier duality 14 OLIVIER WITTENBERG between 1-motives, see [26, §2.4.1]). As a two-term complex of sheaves on Sm/k, the Picard 1-motive 1 0 0 M (U) is canonically isomorphic to [DivX/k,F → PicX/k], where F = X \ U (see Theorem A.4; note 0 0 that the sheaf on Sm/k defined by the Picard variety PicX/k,red is equal to PicX/k). We take the convention that as complexes of sheaves, 1-motives are concentrated in degrees −1 and 0 and written either vertically or between brackets. The natural diagram

0 0 DivX/k,F DivX/k,F

⋆ 0 0 0 Gm RX/k DivX/k PicX/k 0

1 defines a 2-extension of M (U) by Gm in the category C(Sh(Sm/k)) of complexes of étale sheaves ⋆ M on Sm/k, where Gm and RX/k are regarded as complexes concentrated in degree 0. We let E (U) denote this 2-extension, considered as an object of C(C(Sh(Sm/k))). The notation EM(U) is justified by Corollary 4.1.5 below. ′ Recall that if S is a k-scheme, an S-morphism f : U ×k S → U ×k S is said to be universally dominant ′ if its image contains a dense open subset of every fibre of U ×k S → S. For fixed S ∈ Sm/k, we shall now prove (in Proposition 4.1.3 below) that as objects of the category C(C(Sh(Sm/S))), the 2-extensions M E (U)|S are functorial in U with respect to universally dominant S-morphisms, where −|S denotes restriction to Sm/S. Let X′ be another smooth proper geometrically integral variety over k and U ′ ⊆ X′ be a dense ′ open subset. Let S ∈ Sm/k and let f : U ×k S → U ×k S be a universally dominant S-morphism. ⋆ ⋆ Clearly f induces a morphism of sheaves R ′ → R . Let W ⊆ X × S denote the domain X ×kS/S X×kS/S k 99K ′ of definition of the rational map X ×k S X ×k S induced by f. For any T ∈ Sm/S, all points of ′ codimension 1 of X ×k T are contained in W ×S T , since X ×k S → S is proper and X ×k S is normal.

Hence Div(X ×k T/T ) = Div(W ×S T/T ) for all T ∈ Sm/S, and consequently DivX×kS/S = DivW/S. ′ Composing this canonical isomorphism with the morphism DivX ×kS/S → DivW/S induced by f : W → ′ ′ X ×k S yields a morphism DivX ×kS/S → DivX×kS/S. We will refer to it simply as the morphism ⋆ ⋆ induced by f. The two morphisms R ′ → R and Div ′ → Div induced by f X ×kS/S X×kS/S X ×kS/S X×kS/S 1 1 ′ 1 give rise to a natural morphism M (f): M (U )|S → M (U)|S in C(Sh(Sm/S)) and more generally to M ′ M a natural morphism E (U )|S → E (U)|S in C(C(Sh(Sm/S))).

Proposition 4.1.3 — Let X′, X′′ be smooth proper geometrically integral varieties over k and U ′ ⊆ X′, ′′ ′′ ′ ′ ′′ U ⊆ X be dense open subsets. Let S ∈ Sm/k. Let f : U ×k S → U ×k S and g : U ×k S → U ×k S M ′′ M be universally dominant S-morphisms. Then the morphism E (U )|S → E (U)|S induced by g ◦ f is M ′′ M ′ M ′ M equal to the composition of the morphisms E (U )|S → E (U )|S and E (U )|S → E (U)|S induced by g and f.

0 0 Proof — It suﬃces to prove that the morphism of sheaves Div ′′ → Div induced by g ◦ f is X ×kS/S X×kS/S 0 0 0 0 equal to the composition of the morphisms Div ′′ → Div ′ and Div ′ → Div X ×kS/S X ×kS/S X ×kS/S X×kS/S induced by g and f. Lest the reader think that such an assertion must be trivial, we point out that the corresponding statement with Div instead of Div0 is clearly false, even when S = Spec(k) (for instance 2 ′ ′′ 2 ′ 2 ′′ choose a point P ∈ P (k), let U = U = U = Pk \ {P }, f = g = Id, X = Pk, and let X and X be 2 the blowing-up of P in Pk). ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 15

0 ′′ 0 It is even enough to prove that for any T ∈ Sm/S, the map Div (X ×k T/T ) → Div (X ×k T/T ) induced by g◦f is equal to the composition of the maps induced by g and f. Furthermore we may assume that T = S = Spec(k), by extending scalars from k to the function fields of the connected components of T . Indeed, if T is connected, there is for Y ∈ Sm/T a canonical inclusion Div(Y ×k T/T ) ⊆ Div(Y ×k η/η) which is functorial in Y , where η denotes the generic point of T . Finally, we may assume that k is algebraically closed. We are now reduced to showing that if f : X 99K X′ denotes a dominant rational map between smooth proper varieties over an algebraically closed field k, and W ⊆ X is its domain of definition, then the formation of the morphism Div0(X′) → Div0(W ) = Div0(X) induced by f is compatible with composition. Let U ⊆ W and U ′ ⊆ X′ be dense open subsets such that f(U) ⊆ U ′. Denote ′ ′ ′ 0 0 ′ F = X \ U and F = X \ U . The morphism AlbU/k → AlbU ′/k determined by f : U → U induces a 0 0 morphism from the character group of TAlbU ′/k to the character group of TAlbU/k. These groups are 0 ′ 0 canonically isomorphic to DivF ′ (X ) and DivF (X) respectively (Theorem A.4). It is straightforward 0 ′ 0 to check that the morphism DivF ′ (X ) → DivF (X) thus obtained coincides with the restriction of the 0 ′ 0 0 0 morphism Div (X ) → Div (X) induced by f. Now the formation of the morphism TAlbU/k → TAlbU ′/k determined by f is clearly compatible with composition. This remark concludes the proof, thanks to 0 0 0 ′ 0 ′ ′ the formulas Div (X) = lim Div (X) and Div (X ) = lim Div ′ (X ), where F (resp. F ) ranges over −→ F −→ F all closed subsets of X (resp. X′) of codimension 1.

Corollary 4.1.4 — In the situation of Proposition 4.1.3, if f is an isomorphism, then the morphism M ′ M E (U )|S → E (U)|S induced by f is an isomorphism. On taking S = Spec(k), U ′ = U, and f = Id in Corollary 4.1.4, we obtain:

Corollary 4.1.5 — The 2-extension EM(U) depends on U but does not depend on the smooth com- pactiﬁcation U ⊆ X (up to a canonical isomorphism in C(C(Sh(Sm/k)))).

In view of Corollary 4.1.5, we may omit to specify a smooth compactification of U when talking about EM(U), provided that at least one such compactification exists. For future use, we note that torsors under semi-abelian varieties are known to admit smooth compactifications over arbitrary fields. This follows from the existence of smooth equivariant compactifications of tori (for which see [4]). The 2-extensions EM(−) are also functorial in a weaker sense with respect to morphisms which are not necessarily dominant:

Proposition 4.1.6 — Let f : V → W be a morphism of smooth geometrically integral k-varieties which admit smooth compactiﬁcations. The 2-extension EM(W ), regarded as a 2-extension in the abelian category C(Sh(Sm/k)), is canonically Yoneda equivalent to the pullback of EM(V ) by the morphism M 1(W ) → M 1(V ) induced by f.

0 Proof — Let V ⊆ Y and W ⊆ Z denote smooth compactiﬁcations of V and W . Let D ⊆ DivZ/k be 0 ⋆ ⋆ the smallest subsheaf in groups containing DivZ/k,Z\{f(v)} for all closed points v ∈ V . Let R ⊆ RZ/k ⋆ 0 denote the inverse image of D by the natural map RZ/k → DivZ/k, and let H be the complex of sheaves 0 [DivZ/k,Z\W → D] with D in degree 0. By choosing a closed point of V and using the fact that the Picard group of a local ring vanishes, one sees that the natural complex

⋆ 1 (4.1.2) 0 Gm R H M (W ) 0 16 OLIVIER WITTENBERG is an exact sequence in the abelian category C(Sh(Sm/k)). The rational map Y 99K Z induced by f is deﬁned on an open subset Y ′ ⊆ Y whose complement has codimension ≥ 2 in Y . Therefore we 0 0 ⋆ ⋆ can deﬁne natural pullback morphisms D → DivY ′/k = DivY/k and R → RY/k, and hence a natural morphism of exact sequences from (4.1.2) to EM(V ) whose leftmost (resp. rightmost) component is the 1 1 1 identity of Gm (resp. is the morphism M (f): M (W ) → M (V ) induced by f). As a consequence, the 2-extension (4.1.2) is canonically Yoneda equivalent to the pullback of EM(V ) by M 1(f). Now there is a natural morphism of exact sequences from (4.1.2) to EM(W ) whose leftmost (resp. rightmost) 1 M component is the identity of Gm (resp. of M (W )), so the 2-extensions E (W ) and (4.1.2) are also canonically Yoneda equivalent.

Let us now turn our attention to the classes of E(X) and of EM(U) in the corresponding groups of extensions. For any S ∈ Sm/k, any 2-extension

f 1 (4.1.3) 0 Gm,S E F M (U)|S 0

1 in C(Sh(Sm/S)) induces a canonical quasi-isomorphism C(Gm,S[1] → C(f)) → M (U)|S, where C(−) denotes the mapping cone of a morphism, and therefore a distinguished triangle

1 (4.1.4) Gm,S[1] C(f) M (U)|S Gm,S[2] in the bounded derived category Db(Sh(Sm/S)) if E and F are bounded. We shall speak of the class in the hyperext group 2 1 1 ExtSh(Sm/S)(M (U)|S , Gm,S) = HomDb(Sh(Sm/S))(M (U)|S , Gm,S[2]) 1 determined by the 2-extension (4.1.3) to refer to the morphism M (U)|S → Gm,S[2] which appears in the exact triangle (4.1.4) associated with that 2-extension.

M 2 1 Proposition 4.1.7 — The order of the class of E (U) in the hyperext group ExtSh(Sm/k)(M (U), Gm) 2 divides the order of the class of E(X) in ExtSh(Sm/k)(PicX/k, Gm). Let k denote a separable closure of k.

If NS(U ⊗k k) = 0, then these orders are equal.

Proof — Let Y ∈ C(Sh(Sm/k)) denote the complex [DivX/k,F → PicX/k], where PicX/k is placed in degree 0, and let B denote the 2-extension of Y by Gm deﬁned by the diagram

DivX/k,F DivX/k,F

⋆ 0 Gm RX/k DivX/k PicX/k 0. The natural exact sequence

0 PicX/k Y DivX/k,F [1] 0 in C(Sh(Sm/k)) induces a distinguished triangle

PicX/k Y DivX/k,F [1] PicX/k[1] ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 17

b in D (Sh(Sm/k)). Let us apply the functor Hom(−, Gm[2]) to it. We obtain an exact sequence

1 2 2 ExtSh(Sm/k)(DivX/k,F , Gm) ExtSh(Sm/k)(Y, Gm) ExtSh(Sm/k)(PicX/k, Gm).

Its first term vanishes, by Shapiro’s lemma and Hilbert’s Theorem 90. Consequently the rightmost map of this exact sequence is injective. This map sends the class of B to the class of E(X), hence these two classes have the same order. Now there is a natural morphism r : M 1(U) → Y , and the class of EM(U) 2 2 2 1 is the image of the class of B by the map Ext (r, Gm): ExtSh(Sm/k)(Y, Gm) → ExtSh(Sm/k)(M (U), Gm) it induces. The first assertion of Proposition 4.1.7 is thus proved; moreover, to establish the second 2 assertion, it suffices to show that Ext (r, Gm) is bĳective as soon as NS(U ⊗k k) = 0. Let NSX/k ∈ 0 Sh(Sm/k) denote the quotient PicX/k/PicX/k. If NS(U ⊗k k) = 0, the natural morphism of étale sheaves DivX/k,F → NSX/k is surjective. It follows that r is a quasi-isomorphism, and hence indeed 2 that Ext (r, Gm) is bĳective. 4.2. Main construction. Before describing the construction alluded to in the introduction of §4, we briefly sketch, in the particular case where U = X and X is a torsor under an abelian variety A over k, a construction which exploits the same idea while avoiding most of the technical details. Here the goal is to reconstruct X from EM(X) alone; note that the 2-extension EM(X) is simply the pullback of E(X) 0 by the natural map PicX/k → PicX/k since X is proper. Let S ∈ Sm/k. Let AS denote the category 0 whose objects are the 2-extensions of PicX/k ×k S by Gm,S in Sh(Sm/S), and whose morphisms are 0 the morphisms of complexes which induce the identity on PicX/k ×k S and on Gm,S. Since this is only a sketch, we ignore set-theoretic problems and pretend that AS is a small category (one should really fix a large enough cardinal κ and work only with sheaves which take values in abelian groups of cardinality ≤ κ). We can then consider the geometric realisation RS of the nerve of AS. This is a topological space. Objects of AS define points of RS, and connected components of RS correspond to Yoneda equivalence classes of 2-extensions. Let π(S) denote the set of homotopy classes of paths on RS M ∅ M from the trivial 2-extension to E (X)|S . Note that π(S) 6= if and only if the class of E (X)|S 2 0 in ExtSh(Sm/S)(PicX/k ×k S, Gm,S) is 0; in a sense, π(S) is the set of “Yoneda trivialisations” of the M 2-extension E (X)|S up to a certain homotopy relation. Clearly π(S) depends functorially on S, and thus defines a presheaf in sets on Sm/k. Let π be the associated étale sheaf. Then we claim that π and X are canonically isomorphic as étale sheavese on Sm/k. To prove it one first reduces to establishinge that if X = A, a certain canonical morphism α: A → π is an isomorphism. One then identifies π(S) with the fundamental group of RS, where the triviale 2-extensione is chosen as base point. According to Retakh [27, 1 0 Theorem 1], this fundamental group is canonically isomorphic to ExtSh(Sm/S)(PicX/k ×k S, Gm,S), which 0 0 in turn is canonically isomorphic to A(S) by the Barsotti-Weil formula (note that PicX/k = PicA/k). It only remains to be shown that the resulting canonical isomorphism A(S) −→∼ π(S) coincides with α, and this is a consequence of the independent results of §4.3. e This construction relies on the description of Extn groups in abelian categories in terms of n-extensions and Yoneda equivalence. To carry it out in the general case, one would need a similar description for 2 1 the hyperext group ExtSh(Sm/S)(M (U)|S, Gm,S) as well as a version of Retakh’s result which could be applied in this context. As it turns out, both of these tools are contained in the paper of Neeman and Retakh [23]; however, making explicit the construction given in [23] of a canonical isomorphism between the fundamental group of the nerve of a category analogous to AS and the hyperext group 1 1 ExtSh(Sm/S)(M (U)|S , Gm,S), and then using it to prove that α is an isomorphism, is an excessively e 18 OLIVIER WITTENBERG daunting task. Fortunately, a shortcut using derived categories is available, as was explained to the author by Shoham Shamir (see Lemma 4.4.1). In the proofs below, we thus do not have recourse to the results of [27] or [23], at the expense of employing a more abstract definition for the sheaf π.

e1 Construction 4.2.1 — Let G be a semi-abelian variety over k (so that M1(G)= G and M (G) is the 1-motive dual to G). To every 2-extension

1 (4.2.1) 0 Gm E F M (G) 0 1 of M (G) by Gm in C(Sh(Sm/k)), where E is concentrated in degree 0 and F in degrees −1 and 0, we are going to associate a k-torsor under G.

The conditions on E and F are not necessary to carry out the construction, but they make some of the arguments simpler and are unrestrictive enough for our purposes. Let us ﬁrst deﬁne a category AS(G) for any S ∈ Sm/k. Its objects are the 2-extensions

f 1 (4.2.2) 0 Gm,S E F M (G)|S 0 in C(Sh(Sm/S)) such that E is concentrated in degree 0 and F in degrees −1 and 0. We deﬁne a morphism from (4.2.2) to

′ ′ f ′ 1 (4.2.3) 0 Gm,S E F M (G)|S 0

′ b in the category AS(G) to be a morphism C(f) → C(f ) in D (Sh(Sm/S)) such that the diagram 1 Gm,S[1] C(f) M (G)|S Gm,S[2] (4.2.4)

′ 1 Gm,S[1] C(f ) M (G)|S Gm,S[2] commutes, where the rows are the distinguished triangles (4.1.4) associated with (4.2.2) and (4.2.3). We note that all morphisms in AS(G) are isomorphisms, by the triangulated ﬁve lemma ([14, Cor. 1.5.5]). Furthermore, we note that two objects of AS(G) determine the same element of the hyper- 2 1 ext group ExtSh(Sm/S)(M (G)|S , Gm,S) if and only if there exists a morphism between them in AS(G).

A For any u, v ∈ Spec(k)(G) and any S ∈ Sm/k, we let πu,v(S) = HomAS (G)(u|S, v|S ). This defines a presheaf in sets πu,v on Sm/k. Let πu,v denote the associated étale sheaf. If u = v, then composition defines a group structure on πu,v. e We now consider the casee u = v = EM(G) in more detail. For any S ∈ Sm/k and any g ∈ M G(S), translation by g on G defines an automorphism of E (G)|S in C(C(Sh(Sm/S))), according to M 1 Corollary 4.1.4. This automorphism of E (G)|S induces the identity on M (G)|S , so that it determines an element of πEM(G),EM(G)(S). Hence we obtain a morphism of presheaves αG : G → πEM(G),EM(G) on Sm/k. We let αG : G → πEM(G),EM(G) denote the associated morphism of étale sheaves. Clearly it is a morphism of sheavese ine groups.

Theorem 4.2.2 — The morphism αG is an isomorphism. e The proof of Theorem 4.2.2 is quite involved; we defer it to §4.4. ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 19

We are ﬁnally in a position to describe Construction 4.2.1. Let u ∈ ASpec(k)(G). For any S ∈ Sm/k, the action of πEM(G),EM(G)(S) on πEM(G),u(S) by composition is simply transitive. As a consequence, the sheaf πEM(G),u is a torsor under the sheaf in groups πEM(G),EM(G). By Theorem 4.2.2, we can therefore regarde it via αG as a torsor under the sheaf deﬁnede by G. Now every étale sheaf on Sm/k which is a torsor under Ge is representable by a scheme (by Galois descent, see [2, 6.5/1]); we thus obtain a k-torsor under G, and this is the torsor that Construction 4.2.1 associates to u. We now state and prove the main theorem of this section.

Theorem 4.2.3 — Let X be a smooth proper geometrically integral variety over a ﬁeld k and U ⊆ X 0 M be a dense open subset. Then the k-torsor under AlbU/k associated to E (U) by Construction 4.2.1 is 1 canonically isomorphic to AlbU/k.

1 1 1 1 Proof — The morphism of 1-motives M (uU ): M (AlbU/k) → M (U) induced by the canonical mor- 1 1 phism uU : U → AlbU/k is an isomorphism, since its Cartier dual M1(uU ): M1(U) → M1(AlbU/k) is M M 1 clearly an isomorphism. Therefore it follows from Proposition 4.1.6 that E (U) and E (AlbU/k) are 0 canonically isomorphic in ASpec(k)(AlbU/k), and hence that the k-torsors associated to them by Con- struction 4.2.1 are canonically isomorphic. As a consequence, in order to prove Theorem 4.2.3, we may 1 0 assume U = AlbU/k. Let G = AlbU/k. For any S ∈ Sm/k, an element of U(S) canonically determines an ∼ M ∼ M S-isomorphism U ×k S −→ G×k S, hence, by Corollary 4.1.4, an isomorphism E (G)|S −→ E (U)|S in C(C(Sh(Sm/S))), hence an element of πEM(G),EM(U)(S). We thus obtain a morphism of étale sheaves of sets αU : U → πEM(G),EM(U). It only remains to be shown that αU is an isomorphism. This question is locale for the étalee topology on Sm/k, so that by extending scalarse and choosing a point of U, we may assume that U = G. In this case αU = −αG; applying Theorem 4.2.2 concludes the proof. e e Corollary 4.2.4 — Let X be a smooth proper geometrically integral variety over a ﬁeld k. For any dense M 2 1 open subset U ⊆ X, the order of the class of E (U) in the hyperext group ExtSh(Sm/k)(M (U), Gm) is 2 equal to P (U). The order of the class of E(X) in ExtSh(Sm/k)(PicX/k, Gm) is equal to Pgen(X).

Proof — The second assertion of the corollary follows from the first and from Proposition 4.1.7, so we need only prove the first assertion. We fix a dense open subset U ⊆ X and an integer n ≥ 1, n 1 and let f : U → AlbU/k denote the composition of uU : U → AlbU/k with the canonical morphism 1 n n AlbU/k → AlbU/k. Then M1(f): M1(U) → M1(AlbU/k) identifies with multiplication by n on M1(U) n 1 1 n 1 via the canonical isomorphism M1(AlbU/k)= M1(U); therefore the map M (f): M (AlbU/k) → M (U) identifies with multiplication by n on M 1(U). From this we deduce that the class of EM(U) in 2 1 M n ExtSh(Sm/k)(M (U), Gm) is killed by n if and only if the class of E (AlbU/k) is trivial, thanks to M n Proposition 4.1.6. The corollary will now be established if we prove that the class of E (AlbU/k) n ∅ n ∅ is trivial if and only if AlbU/k(k) 6= . One implication is easy: if AlbU/k(k) 6= , then Proposi- n M n tion 4.1.6 applied to a k-morphism Spec(k) → AlbU/k shows that the class of E (AlbU/k) is trivial. M n 0 ∅ Let us now assume, conversely, that the class of E (AlbU/k) is trivial. We have AlbU/k(k) 6= , M 0 M 0 hence, again by Proposition 4.1.6, the class of E (AlbU/k) is trivial. In particular E (AlbU/k) and M n 2 1 E (AlbU/k) define the same class in the hyperext group ExtSh(Sm/k)(M (U), Gm). It follows that 20 OLIVIER WITTENBERG

M n π M 0 M n (k) 6= ∅, which in turn implies that the k-torsor associated to E (Alb ) by E (AlbU/k),E (AlbU/k) U/k n Construction 4.2.1 has a rational point. By Theorem 4.2.3, this torsor is isomorphic to AlbU/k; hence n ∅ AlbU/k(k) 6= , and the corollary is proved.

4.3. Explicit Poincaré sheaves on abelian varieties. Let k be a field and A be an abelian variety P 0 over k. The Poincaré sheaf A is an invertible sheaf on A ×k PicA/k which plays a fundamental rôle in the duality theory of abelian varieties. One can apply the universal property of the dual abelian variety P 0 0 0 to A in two ways, thereby producing a morphism A → Pic 0 and a morphism PicA/k → PicA/k. PicA/k/k P The latter is the identity, by definition of A, and the former is the biduality isomorphism [22, p. 132]. Our goal in this subsection is to exhibit an explicit Poincaré sheaf on any abelian variety. As a consequence we obtain a very explicit description of the biduality isomorphism. These results will be used in §4.4 to prove Theorem 4.2.2. We consider A ×k A as an A-scheme via the second projection. Let ∆ ⊆ A ×k A denote the diagonal and 0 ⊆ A ×k A denote the zero section of the second projection (so that both ∆ and 0 are isomorphic to A as k-schemes). Let V = (A ×k A) \ (∆ ∪ 0). Let d : R⋆ −→ Div0 × G A A×kA/A,V A×kA/A,V m,A be the morphism of étale sheaves on Sm/A defined by r 7→ (div(r), r(0)/r(∆)).

Lemma 4.3.1 — The natural sequence of étale sheaves on Sm/A

(4.3.1) 0 G Coker(d ) Pic0 0, m,A A A×kA/A where the ﬁrst map is deﬁned by x 7→ (0,x) and the second map sends the class of (d, x) in Coker(dA) to the linear equivalence class of d, is exact.

Proof — Only exactness on the right requires a proof. It suﬃces to check that for any T ∈ Sm/A, any t ∈ T , and any D ∈ Div(A ×k T ), there is an open subset U of T containing t such that the restriction of D to A ×k U is linearly equivalent, on A ×k U, to a divisor whose support is contained in V ×A U; and this follows from the vanishing of the Picard group of the semilocal ring of A ×k T at the two points 0 × t and s(t) × t, where s: T → A denotes the structural morphism of T (the Picard group of any semilocal ring vanishes).

The exact sequence (4.3.1) deﬁnes a torsor under G over Pic0 (whose sections over an open m A×kA/A subset U of Pic0 are the elements of Coker(d )(U) which lift the inclusion U ⊆ Pic0 seen A×kA/A A A×kA/A as an element of Pic0 (U); note that U ∈ Sm/A), hence an invertible sheaf Q on Pic0 = A×kA/A A A×kA/A 0 PicA/k ×k A. To identify line bundles, invertible sheaves, and torsors under Gm, we use covariant conventions: the line bundle associated with an invertible sheaf L on a scheme S is the (relative) spectrum of the L symmetric algebra of the dual of ; the torsor under Gm associated with a line bundle L → S is the complement of the zero section in L, with the induced action of Gm.

Q P Theorem 4.3.2 — The invertible sheaf A is isomorphic to the Poincaré sheaf A. ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 21

0 Corollary 4.3.3 — The biduality isomorphism A → Pic 0 sends a point a ∈ A(k) to the class of PicA/k/k 0 the invertible sheaf on PicA/k deﬁned by the exact sequence of étale sheaves on Sm/k

⋆ 0 0 (4.3.2) 0 Gm Coker(RA/k,A\{a,0} → DivA/k,A\{a,0} × Gm) PicA/k 0,

⋆ 0 where the map RA/k,A\{a,0} → DivA/k,A\{a,0} × Gm is deﬁned by r 7→ (div(r), r(0)/r(a)).

Proof of Theorem 4.3.2 — For a = 0, the exact sequence (4.3.2) is clearly split. In other words, the morphism A → Pic 0 which corresponds to Q by the universal property of the Picard functor PicA/k/k A 0 maps 0 to 0. Hence it induces a morphism of abelian varieties A → Pic 0 . Let σA ∈ Endk(A) PicA/k/k ∼ 0 denote the composition of this morphism with the inverse of the biduality isomorphism A −→ Pic 0 . PicA/k/k P Q The assertion that A and A are isomorphic now amounts to the equality σA = Id. There exist a Jacobian variety J over k and a smooth surjective morphism m: J → A ([9, Corol- Q Q 0 lary 2.5]). It is straightforward to check that the pullbacks of A and J to PicA/k ×k J are isomorphic (note that J ∈ Sm/A). As a consequence, the square

J m A

σJ σA J m A commutes; hence σJ = Id implies σA = Id. We can therefore assume that A is a Jacobian variety, i.e., that there exists a smooth proper geo- 0 metrically connected curve C over k such that A = PicC/k. After replacing k with a finite separable extension (which, for the purpose of proving the equality of ∅ endomorphisms σA = Id, is harmless), we may assume that C(k) 6= . Fix a point c0 ∈ C(k). There is a canonical embedding ι: C → A such that ι(c0) = 0 ([21, §2]). In order to prove that σA = Id, it is enough to prove that σA ◦ ι = ι, since the image of ι generates A. As C is reduced, it even suffices to establish that σA(ι(c)) = ι(c) for every closed point c ∈ C whose residue field is separable over k, or equivalently that P | 0 and Q | 0 are isomorphic invertible sheaves for every A PicA/k×{ι(c)} A PicA/k×{ι(c)} such c ∈ C. Fix a closed point c ∈ C with separable residue field over k. After replacing k with the 0 0 residue field of c, we may assume that c ∈ C(k). The morphism PicA/k → PicC/k = A induced by ι ′ 0 is an isomorphism ([21, §6]). Let ι : C → PicA/k denote the composition of ι with the inverse of this 0 isomorphism. Again by [21, §6], two algebraically equivalent invertible sheaves on PicA/k are isomorphic ′ ⋆P if and only if their inverse images by ι are isomorphic. In particular, we need only prove that s A ⋆Q 0 ′ and s A are isomorphic, where s: C → PicA/k ×k A is the morphism defined by s(x) = (ι (x), ι(c)). ⋆P O According to [21, §6, 6.11], the invertible sheaf s A is isomorphic to (c − c0). Let X → C and O Y → C denote the torsors under Gm respectively defined by the invertible sheaves (c − c0) and ⋆Q s A. The embedding ι induces a morphism from the exact sequence (4.3.2) with a = ι(c) to the exact sequence

d 0 G Coker R⋆ −−−→C Div0 × G Pic0 0, m C/k,C\{c,c0} C/k,C\{c,c0} m C/k 22 OLIVIER WITTENBERG where dC is defined by r 7→ (div(r), r(c0)/r(c)). This is an isomorphism of exact sequences, by the five lemma. Hence the torsor Y → C can be described as follows: its sections over an open sub- 0 set U ⊆ C are the liftings in Coker(dC )(U) of the element of PicC/k(U) defined by the composition ι 0 U ⊆ C −→ A = PicC/k (note that U ∈ Sm/k). The torsor X → C is easier to describe: its sections over ⋆ an open subset U ⊆ C are the rational functions g ∈ k(C) such that the divisor div(g)+ c − c0 is supported on C \ U. We shall now exhibit a Gm-equivariant C-morphism ϕ: X → Y . Let us fix once and for all a ⋆ rational function π ∈ k(C) which is a uniformiser at c0 and which moreover is invertible at c if c 6= c0. (The existence of π follows from the vanishing of the Picard group of the semilocal ring of C at c and c0.) We define ϕ in the neighbourhood of an arbitrary closed point c1 ∈ C. Let U ⊆ C be an open subset containing c1. Let D ∈ Div(C ×k U) be a divisor linearly equivalent to ∆ − (c0 × U) where ∆ denotes the image of the diagonal embedding U ֒→ C ×k U), with disjoint support from) (c × c1) ∪ (c0 × c1). Such a divisor exists because the Picard group of a semilocal ring is trivial. By shrinking U if necessary, we may assume that D has support disjoint from (c × U) ∪ (c0 × U). Let ⋆ ′ ⋆ h ∈ k(C×k U) satisfy div(h)=∆−(c0 ×U)−D, and let h ∈ k(C×k U) be the rational function defined ′ ′ by h (x,y)= h(x,y)π(x). The divisor of h coincides with ∆ in a neighbourhood of (c × U) ∪ (c0 × U). ′ ′ ⋆ As a consequence, the rational function y 7→ g(y)h (c, y)/h (c0,y) is invertible on U for any g ∈ k(C) such that div(g)+ c − c0 is supported on C \ U. We let ϕ(U): X(U) → Y (U) be the map which sends g to the class in Coker(dC )(U) of the pair consisting of the divisor D ∈ Div(C ×k U) and the function ′ ′ (y 7→ g(y)h (c, y)/h (c0,y)) ∈ Gm(U). Note that the restrictions of D to the geometric fibres of the second projection C ×k U → U are indeed algebraically equivalent to 0 and supported on C \ {c, c0}. ⋆ Suppose D1 ∈ Div(C ×k U) and h1 ∈ k(C ×k U) are another divisor and another rational function satisfying the same requirements as D and h, i.e., D1 is linearly equivalent to ∆ − (c0 × U), has support ⋆ disjoint from (c × U) ∪ (c0 × U), and satisfies div(h1)=∆ − (c0 × U) − D1. Then, for any g ∈ k(C) such ′ ′ ′ ′ that div(g)+c−c0 is supported on C\U, the pairs (D,gh (c, −)/h (c0, −)) and (D1, gh1(c, −)/h1(c0, −)), ′ where h1(x,y) = h1(x,y)π(x), define the same class in Coker(dC )(U). Indeed, letting r = h/h1, these two pairs differ by (div(r), r(c0, −)/r(c, −)), which by definition of dC vanishes in Coker(dC )(U). Hence the map ϕ(U) does not depend on the choices of D and h. In addition, it is clearly Gm(U)-equivariant and functorial with respect to U. We have therefore defined a Gm-equivariant morphism ϕ: X → Y in the category of Zariski sheaves of sets on C. Now by Grothendieck’s Hilbert 90 theorem, giving a Gm-equivariant C-morphism between torsors under Gm over C is equivalent to giving a Gm-equivariant morphism between the Zariski sheaves on C they define (see [7, Arcata, II-4]). Moreover, such a morphism is necessarily an isomorphism; hence the proof of Theorem 4.3.2 is complete.

4.4. Proof of Theorem 4.2.2. We now show that the morphism of sheaves αG is an isomorphism. This question is local for the étale topology on Sm/k. By replacing k with a ﬁnitee separable extension, we can therefore assume (for simplicity) that G is an extension of an abelian variety by a split torus. The following lemma and its proof are a variant for the hyperext groups under consideration of a general result on Ext groups in abelian categories which was communicated to the author by Shoham Shamir.

1 Lemma 4.4.1 — Fix S ∈ Sm/k and u ∈ AS(G). For ϕ ∈ HomDb(Sh(Sm/S))(M (G)|S , Gm,S[1]), we 1 ϕ let A(ϕ) denote the composition C(f) → M (G)|S −→ Gm,S[1] → C(f), where the unlabeled maps are taken from the exact triangle (4.1.4) associated with u. Then ϕ 7→ Id + A(ϕ) deﬁnes an isomorphism 1 1 from the hyperext group ExtSh(Sm/S)(M (G)|S , Gm,S) to the group of automorphisms of u in AS(G). ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 23

Proof — The subscript in HomDb(Sh(Sm/S))(−, −) will be dropped for the rest of the proof. Let us denote 1 by ρ: Gm,S[1] → C(f) and σ : C(f) → M (G)|S the maps which appear in the exact triangle (4.1.4) associated with u, so that A(ϕ) = ρ ◦ ϕ ◦ σ. First it is immediate that Id + A(ϕ) is an endomorphism of u in AS(G), and hence an automorphism. We have A(ϕ) ◦ A(ψ) = 0 and A(ϕ + ψ) = A(ϕ)+ A(ψ) for any ϕ and ψ, therefore ϕ 7→ Id+ A(ϕ) is a group morphism. We note that Hom(Gm,S[2],C(f)) = 0 1 1 and Hom(M (G)|S , M (G)|S [−1]) = 0. Indeed the ﬁrst group vanishes because C(f) is concentrated in degrees −1 and 0 (quite generally one has Hom(E, F ) = 0 if E is concentrated in degrees ≤ n 1 and F in degrees ≥ n + 1, for some n ∈ Z); similarly, M (G)|S being concentrated in degrees −1 and 0 implies that the second group injects into the group of homomorphisms from an S-abelian scheme to an S-lattice, hence is trivial. Taking these remarks into account while applying the functors Hom(−,C(f)) 1 and Hom(M (G)|S , −) to the distinguished triangle (4.1.4), we obtain the two exact sequences

1 −◦σ −◦ρ (4.4.1) 0 Hom(M (G)|S ,C(f)) Hom(C(f),C(f)) Hom(Gm,S[1],C(f)) and

1 ρ◦− 1 σ◦− 1 1 (4.4.2) 0 Hom(M (G)|S , Gm,S[1]) Hom(M (G)|S ,C(f)) Hom(M (G)|S , M (G)|S ).

1 It follows that ϕ 7→ A(ϕ) induces a bĳection from Hom(M (G)|S , Gm,S[1]) to the group of morphisms a: C(f) → C(f) such that a ◦ ρ = 0 and σ ◦ a = 0. Now a morphism a: C(f) → C(f) satisfies these conditions if and only if Id + a is an automorphism of u in AS(G), hence the lemma. Let X denote a smooth compactification of G and F = X \ G. For S ∈ Sm/k, we denote by f : R⋆ → [Div0 → Div0 ] the restriction to Sm/S of the morphism of complexes X×kS/S X×kS/S,F ×kS X×kS/S of étale sheaves on Sm/k which appears in the definition of EM(G) (see §4.1). Let 1 1 ∼ βG : ExtSh(Sm/S)(M (G)|S , Gm,S) −−→ πEM(G),EM(G)(S) be the isomorphism given by Lemma 4.4.1. By abuse of notation we shall write αG for the map αG(S): G(S) → πEM(G),EM(G)(S) induced by αG.

Lemma 4.4.2 — Let S ∈ Sm/k. Let G′ be another semi-abelian variety over k and m: G → G′ be a homomorphism. The square

G(S) m G′(S)

−1 −1 β ◦α ′ βG ◦αG G′ G

1 1 Ext (M (m),Gm,S ) 1 1 Sh(Sm/S) 1 1 ′ ExtSh(Sm/S)(M (G)|S , Gm,S) ExtSh(Sm/S)(M (G )|S, Gm,S) commutes.

Proof — We fix a smooth compactification X′ of G′ and let ′ ⋆ 0 0 f : R ′ −→ [Div ′ ′ → Div ′ ] X ×kS/S X ×kS/S,F ×kS X ×kS/S be the restriction to Sm/S of the morphism of complexes which appears in the definition of EM(G′), where F ′ = X′ \G′. Let g ∈ G(S). Translation by g (resp. by m(g)) on G (resp. on G′) induces an auto- ′ b ⋆ ⋆ morphism of the mapping cone C(f) (resp. C(f )) in D (Sh(Sm/S)), which we denote tg (resp. tm(g)). 24 OLIVIER WITTENBERG

′ ′ 1 ′ ′ 1 ′ Let ρ: Gm,S[1] → C(f), ρ : Gm,S[1] → C(f ), σ : C(f) → M (G)|S , and σ : C(f ) → M (G )|S denote M M ′ the maps defined by the exact triangles (4.1.4) associated with E (G)|S and E (G )|S. 1 ′ 1 ′ By Lemma 4.4.1, there exist unique morphisms ϕ: M (G)|S → Gm,S[1] and ϕ : M (G )|S → Gm,S[1] b ⋆ ′ ′ ′ ⋆ ⋆ ′ in D (Sh(Sm/S)) such that ρ◦ϕ◦σ = tg −Id and ρ ◦ϕ ◦σ = tm(g) −Id. We denote by m : C(f ) → C(f) 1 1 ′ 1 b and M (m): M (G )|S → M (G)|S the morphisms in D (Sh(Sm/S)) induced by m (the first one being given by Proposition 4.1.6). The conclusion of Lemma 4.4.2 amounts to the equality ϕ ◦ M 1(m)= ϕ′ in 1 ′ ⋆ ⋆ ⋆ ⋆ ⋆ ′ 1 ′ HomDb(Sh(Sm/S))(M (G )|S, Gm,S[1]). Clearly one has tg ◦m = m ◦tm(g), m ◦ρ = ρ, and M (m)◦σ = σ ◦ m⋆. Hence ρ ◦ ϕ ◦ M 1(m) ◦ σ′ = ρ ◦ ϕ′ ◦ σ′. We now argue as in the proof of Lemma 4.4.1 to deduce from this equality first that ρ ◦ ϕ ◦ M 1(m) = ρ ◦ ϕ′ (using the exact sequence obtained from (4.4.1) by replacing the second occurrence of C(f) with C(f ′), and G, σ, ρ with G′, σ′, ρ′), and then that ϕ ◦ M 1(m)= ϕ′ (using the exact sequence obtained from (4.4.2) by replacing the first three occurrences of G with G′).

−1 The morphism βG ◦ αG depends functorially on S, and can therefore be regarded as a morphism of presheaves. By considering the associated morphism of sheaves, one deduces from Lemma 4.4.2 and from the five lemma that in order to prove that αG is an isomorphism, it is harmless to assume that G is either a split torus or an abelian variety, ande then that G is either Gm or an abelian variety (and 1 accordingly X = Pk or X = G). Under this assumption, we shall establish that the morphism of presheaves αG is itself an isomorphism. 1 0 We first suppose G is an abelian variety, so that M (G) = PicG/k. For any S ∈ Sm/k, the Barsotti- Weil morphism 1 0 0 ExtSh(Sm/S)(PicG/k ×k S, Gm,S) −→ Pic 0 (S), PicG/k/k 0 0 which sends an extension of PicG/k ×k S by Gm,S to the class of the torsor under Gm,S over PicG/k ×k S that it defines, is an isomorphism (see [24, III.18]). Composing its inverse with the biduality isomorphism ∼ 0 ∼ 1 1 G(S) −→ Pic 0 (S) yields an isomorphism γG : G(S) −→ ExtSh(Sm/S)(M (G)|S , Gm,S). Clearly PicG/k/k −1 −1 γG ◦ βG ◦ αG depends functorially on S, hence defines a morphism of presheaves G → G on Sm/k. By Yoneda’s lemma, this morphism comes from an endomorphism m of the abelian variety G. For the morphism of presheaves αG to be an isomorphism, it suffices that m = Id. Now two endomorphisms of a smooth algebraic k-group coincide as soon as they coincide on ℓ-points for all finite separable extensions ℓ/k. As a consequence, we will be done if the morphisms of groups G(ℓ) → πEM(G),EM(G)(ℓ) induced by αG and by βG ◦ γG are equal for all finite separable extensions ℓ/k; this is what we shall prove now. After extending the scalars from k to ℓ, we may assume ℓ = k. ⋆ 0 Let g ∈ G(k). Let d: RG/k,G\{g,0} → DivG/k,G\{g,0} × Gm denote the morphism of étale sheaves 1 1 on Sm/k defined by r 7→ (div(r), r(0)/r(g)). By Corollary 4.3.3, the class in ExtSh(Sm/k)(M (G), Gm) of the natural exact sequence

0 (4.4.3) 0 Gm Coker(d) PicG/k 0

0 b is equal to γG(g). Let ϕ: PicG/k → Gm[1] be the morphism in D (Sh(Sm/k)) corresponding to (4.4.3). We keep the notations A, ρ, and σ from the statement and the proof of Lemma 4.4.1 with S = Spec(k). ⋆ It only remains to be shown that Id + A(ϕ) coincides with the morphism tg : C(f) → C(f) induced ⋆ 0 0 by translation by g on G. The complex C(f) is equal to [RG/k → DivG/k] (with DivG/k in degree 0). ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 25

The morphism A(ϕ) ∈ HomDb(Sh(Sm/k))(C(f),C(f)) can be written as a composition of morphisms of complexes and formal inverses of quasi-isomorphisms of complexes in the following way:

⋆ Id ⋆ RG/k 0 Gm Gm RG/k (4.4.4)

0 0 0 DivG/k PicG/k Coker(d) 0 DivG/k . The natural commutative cube

0 Gm

⋆ ⋆ RG/k RG/k,G\{g,0}

0 PicG/k Coker(d),

0 0 DivG/k DivG/k,G\{g,0}

⋆ where the map RG/k,G\{g,0} → Gm is defined by r 7→ r(g)/r(0), enables one to rewrite (4.4.4) as the composition of a single formal inverse of a quasi-isomorphism followed by a single morphism of complexes. Indeed the frontmost and the backmost faces, regarded as leftwards morphisms of vertical complexes, are quasi-isomorphisms. From this explicit description of A(ϕ), one deduces that the mor- ⋆ b phism Id+ A(ϕ) − tg is equal to 0 in D (Sh(Sm/k)) if and only if the morphism of (vertical) complexes ⋆ ⋆ RG/k,G\{g,0} RG/k (4.4.5) Id−t⋆ 0 g 0 DivG/k,G\{g,0} DivG/k, where the top horizontal map sends a rational function r to the rational function r(g)r(x) x 7−→ , r(0)r(x + g) b is equal to 0 in D (Sh(Sm/k)). For any T ∈ Sm/k and any D ∈ Div(G×k T ) whose support is contained in (G \ {g, 0}) ×k T and whose restriction to every geometric fibre of the second projection G ×k T → T 0 ⋆ is algebraically equivalent to 0, the section of PicG/k on T defined by D − tgD is trivial. Indeed, as is well known, the linear equivalence class of a divisor on an abelian variety is invariant under translations if the divisor is algebraically equivalent to 0 (see [22, p. 75]). Therefore there exists (locally on T for ⋆ the Zariski topology) a rational function r on G ×k T whose divisor is D − tgD. Let h(D) be the unique such r which in addition satisfies r(0) = 1. The association of h(D) to D defines a morphism of étale 0 ⋆ sheaves h: DivG/k,G\{g,0} → RG/k on Sm/k. The diagram obtained by adjoining h to the square (4.4.5) is still commutative. As a consequence, the morphism of complexes (4.4.5) is null-homotopic, hence is indeed equal to 0 in Db(Sh(Sm/k)). 1 The case where G = Gm and X = Pk is similar, though substantially easier; we explain it briefly. Let 1 S ∈ Sm/k. We have M (G)|S = Z[1] in C(Sh(Sm/S)), where Z denotes the constant sheaf Z on Sm/S. In order to prove that αG is an isomorphism, it suffices to check that for any g ∈ G(S), the equality 26 OLIVIER WITTENBERG

⋆ 1 tg = Id+ A(ϕ) holds in HomDb(Sh(Sm/S))(C(f),C(f)), where ϕ: M (G)|S → Gm,S[1] now denotes the b morphism in D (Sh(Sm/S)) obtained by shifting the morphism of sheaves Z → Gm,S which maps 1 to g. For this it suﬃces that the morphism of (vertical) complexes

⋆ ⋆ Z ⊕ RP1 Z ⊕ RP1 S /S S/S

Id−t⋆ 0 g 0 DivP1 DivP1 , S /S S /S

n ⋆ where the top horizontal map sends n ⊕ r to 0 ⊕ (g r/tgr) and both vertical maps send n ⊕ r to div(r)+ n(∞− 0), be null-homotopic. It is easy to see that a homotopy is given by the morphism 0 ⋆ ⋆ DivP1 → Z⊕RP1 deﬁned by D 7→ 0⊕(r/tgr), where r is any rational function such that div(r)= D. S /S S /S Thus the proof of Theorem 4.2.2 is complete.

Appendix A Except for the applications given in §3, most of the present paper deals with smooth varieties over an arbitrary field k. Care has been taken not to exclude imperfect fields from our treatment, with global fields of positive characteristic in mind as possible choices for k. To the best of our knowledge, however, the results we use concerning universal morphisms to semi-abelian varieties cannot be found in the literature without the assumption that k be perfect. We provide in this appendix arguments to circumvent the few issues which show up when dealing with imperfect fields.

Theorem A.1 (Serre) — Let k be a ﬁeld and X be a geometrically integral variety over k, endowed with a rational point x ∈ X(k). There exist a semi-abelian variety A over k and a k-morphism u: X → A mapping x to 0, such that for any semi-abelian variety B over k, any k-morphism X → B mapping x to 0 factors uniquely through u.

Let kalg be an algebraic closure of k. Theorem A.1 is due to Serre when k = kalg ([30, Théorème 7]). To prove it in general, we shall follow closely some of the arguments of [30], and then use the fact that Theorem A.1 is true for kalg to shortcut the rest of the proof. While no condition on k is imposed in the statement of Theorem A.1, the case of interest is really that of a separably closed ﬁeld.

Sketch of proof — If A is a semi-abelian variety over k, a k-morphism a: X → A mapping x to 0 is said to be generating if A is the smallest subgroup scheme of A over k through which a factors. It is said to be maximal if for any semi-abelian variety B over k and any k-morphism b: X → B mapping x to 0, any ﬁnite k-homomorphism h: B → A such that h ◦ b = a is an isomorphism.

Lemma A.2 — Let A be a semi-abelian variety over k and a: X → A be a k-morphism mapping x alg alg alg to 0. If a is generating, then a ⊗k k : X ⊗k k → A ⊗k k is generating.

n n n Proof — Let a : X → A be the map (x1,...,xn) 7→ a(xi), where X denotes the n-fold ﬁbred product of X with itself above k. The reasoning of [30, p.P 10-01] (in terms of scheme-theoretic images) shows that a is generating if and only if an is dominant for large n, and this last condition is clearly preserved by extension of scalars from k to kalg. ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 27

Lemma A.3 — Let A be a semi-abelian variety over k and a: X → A be a k-morphism mapping x to 0. There exist a semi-abelian variety B over k, a ﬁnite k-homomorphism h: B → A, and a k-morphism b: X → B mapping x to 0, such that h ◦ b = a and b is maximal.

Proof — Let A′ be the smallest subgroup scheme of A over k through which a factors. Let A′′ denote the reduced closed subscheme of A′ whose underlying topological space is the connected component of 0 in A′. The k-morphism a factors through A′′ since X is reduced and connected. Now A′′ is a subgroup scheme of A′ over k, according to [12, Proposition 3.1]. Hence A′′ = A′, so that A′ is connected and reduced. It now follows from [12, Proposition 3.1] that A′ is smooth over k. Any smooth connected subgroup scheme of a semi-abelian variety is itself a semi-abelian variety. We may therefore replace A with A′, and assume that a is generating. From this point on, the proof of [30, Théorème 1] applies word for word.

The arguments in the second part of the proof of [30, Théorème 2] require no modiﬁcation, except that the reference to [30, Théorème 1] must be replaced with a reference to Lemma A.3 above. They imply that in order to establish Theorem A.1, it suﬃces to show that the dimensions of the semi-abelian varieties B such that there exists a maximal k-morphism b: X → B mapping x to 0 are bounded. alg Let b: X → B be such a morphism. Let u: X ⊗k k → A be the universal morphism given by alg alg alg Theorem A.1 over k (so A is a semi-abelian variety over k ). Let h: A → B ⊗k k be the unique alg alg k -homomorphism such that h ◦ u = b ⊗k k . The k-morphism b is generating since it is maximal. Therefore, by Lemma A.2, the kalg-morphism h ◦ u is generating. It follows that h is surjective, hence dim(B) is bounded by dim(A).

By a standard Galois descent argument, it follows from Theorem A.1 applied to a separable closure 1 0 of k that the Albanese torsor AlbX/k and the Albanese variety AlbX/k exist for geometrically integral varieties X over an arbitrary field k (see §2 for the definitions of these objects, and [8, Theorem 2.1] or [32, Ch. V, §4] for the Galois descent argument). Let X be a smooth proper geometrically integral variety over k and U ⊆ X be a dense open subset. Under the assumption that k = kalg, Serre gave in [29, Théorème 1] an explicit description of the semi- 0 abelian variety AlbU/k in terms of the Picard variety of X and the group of divisors on X which are algebraically equivalent to 0 and supported on X \ U. It turns out that the proof given in [29] applies verbatim when k is only separably closed, thus yielding, after a slight reformulation (which was not possible at the time [29] was written; indeed [29] predates the definition of 1-motives):

Theorem A.4 (Serre) — Let X be a smooth proper geometrically integral variety over a ﬁeld k and U ⊆ X be a dense open subset. Let k be a separable closure of k and D denote the étale k-group scheme 0 0 deﬁned by D(k) = DivX\U (X ⊗k k). The semi-abelian variety AlbU/k is canonically dual to the 1-motive 0 [D → P ], where P = PicX/k,red is the Picard variety of X and D → P is the natural map.

Corollary A.5 — We keep the hypotheses of Theorem A.4. Let K/k be a separable extension (not necessarily algebraic). Then the natural morphism Albn → Albn ⊗ K is an isomorphism for U⊗kK/K U/k k every n ≥ 0.

The separability hypothesis in Corollary A.5 is known to be superfluous in the classical situation where U = X (see, e.g., [12, Théorème 3.3]). However it cannot be removed in general. Indeed, let k be 1 1 any imperfect field, let X = Pk, let P ∈ Ak be a closed point whose residue field is purely inseparable 1 over k, let K be the residue field of P , and let U = Ak \ {P }; then, according to Theorem A.4, the 28 OLIVIER WITTENBERG natural morphism of tori Alb0 → Alb0 ⊗ K is an isogeny of degree [K : k] (more precisely, U⊗kK/K U/k k the map of character groups it induces is isomorphic to multiplication by [K : k] on Z).

References 1. M. Borovoi, J-L. Colliot-Thélène, and A. N. Skorobogatov, The elementary obstruction and homogeneous spaces, to appear in Duke Math. J. 2. S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21, Springer-Verlag, Berlin, 1990. 3. J-L. Colliot-Thélène, Un théorème de finitude pour le groupe de Chow des zéro-cycles d’un groupe algébrique linéaire sur un corps p-adique, Invent. math. 159 (2005), no. 3, 589–606. 4. J-L. Colliot-Thélène, D. Harari, and A. N. Skorobogatov, Compactification équivariante d’un tore (d’après Brylinski et Künnemann), Expo. Math. 23 (2005), no. 2, 161–170. 5. J-L. Colliot-Thélène and J-J. Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492. 6. P. Deligne, Cohomologie à supports propres, Exp. XVII, Théorie des topos et cohomologie étale des schémas. Tome 3, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Lecture Notes in Mathematics, vol. 305. 1 7. , Cohomologie étale, Springer-Verlag, Berlin, 1977, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 2 , avec la collaboration de J-F. Boutot, A. Grothendieck, L. Illusie et J-L. Verdier, Lecture Notes in Mathematics, Vol. 569. 8. D. Eriksson and V. Scharaschkin, On the Brauer-Manin obstruction for zero-cycles on curves, submitted. 9. O. Gabber, On space filling curves and Albanese varieties, Geom. Funct. Anal. 11 (2001), no. 6, 1192–1200. 10. A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, III, Publ. Math. de l’I.H.É.S. (1966), no. 28. 11. , Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): IV. Étude locale des schémas et des morphismes de schémas, IV, Publ. Math. de l’I.H.É.S. (1967), no. 32. 12. , Technique de descente et théorèmes d’existence en géométrie algébrique. VI. Les schémas de Picard: propriétés générales, Exp. No. 236, Séminaire Bourbaki, t. 14, 1961/1962, Soc. Math. France, Paris, 1995, pp. 221–243. 13. D. Harari and T. Szamuely, Local-global principles for 1-motives, submitted. 14. M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292, Springer-Verlag, Berlin, 1994. 15. J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 32, Springer-Verlag, Berlin, 1996. 16. S. Lang and J. Tate, Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80 (1958), 659–684. 17. E. Matlis, Applications of duality, Proc. Amer. Math. Soc. 10 (1959), 659–662. 18. A. S. Merkurjev and I. A. Panin, K-theory of algebraic tori and toric varieties, K-Theory 12 (1997), no. 2, 101–143. 19. J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150. 20. , Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press Inc., Boston, MA, 1986. 21. , Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 167–212. 22. D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. 23. A. Neeman and V. Retakh, Extension categories and their homotopy, Compositio Math. 102 (1996), no. 2, 203–242. 24. F. Oort, Commutative group schemes, Lecture Notes in Mathematics, vol. 15, Springer-Verlag, Berlin, 1966. 25. N. Ramachandran, Duality of Albanese and Picard 1-motives, K-Theory 22 (2001), no. 3, 271–301. 26. M. Raynaud, 1-motifs et monodromie géométrique, Astérisque (1994), no. 223, 295–319, Périodes p-adiques (Bures- sur-Yvette, 1988). 27. V. S. Retakh, Homotopic properties of categories of extensions, Russian Math. Surveys 41 (1986), no. 6, 217–218. 28. J-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math. 327 (1981), 12–80. 29. J-P. Serre, Morphismes universels et différentielles de troisième espèce, Exp. No. 11, Séminaire C. Chevalley, t. 4, 1958/1959, École Normale Supérieure, Paris, 1960. 30. , Morphismes universels et variétés d’Albanese, Exp. No. 10, Séminaire C. Chevalley, t. 4, 1958/1959, École Normale Supérieure, Paris, 1960. ONALBANESETORSORSANDTHEELEMENTARYOBSTRUCTION 29

31. , Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Actualités Sci. Indust., No. 1296. Hermann, Paris, 1962. 32. , Groupes algébriques et corps de classes, Deuxième édition, Publications de l’Institut de Mathématique de l’Université de Nancago, VII, Actualités Sci. Indust., No. 1264. Hermann, Paris, 1975. 33. , Cohomologie galoisienne, Cinquième édition, Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994. 34. A. N. Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. 35. , On the elementary obstruction to the existence of rational points, Mat. Zametki 81 (2007), no. 1, 112–124 (Russian). 36. J. van Hamel, The Brauer-Manin obstruction for zero-cycles on Severi-Brauer ﬁbrations over curves, J. London Math. Soc. (2) 68 (2003), no. 2, 317–337. 37. , Lichtenbaum-Tate duality for varieties over p-adic ﬁelds, J. reine angew. Math. 575 (2004), 101–134.

Department of Mathematics, Rice University, 6100 S. Main St., Houston, TX 77251-1892, USA E-mail address: [email protected]